2.1 Overview

Suppose that $\mathcal {H}$ is a sequence of s-uniform hypergraphs and let $r \geqslant 2$ be an integer. The following function takes centre stage in our considerations:

In order to phrase the five assumptions on the sequence $\mathcal {H}$ that guarantee that non-r-colourability has a sharp threshold in $\mathcal {H}_p$ , we need to introduce three simple notions. A star in $\mathcal {H}$ is a collection of $r-1$ edges that pairwise intersect in a single vertex called the centre of the star. A constellation is a collection of s disjoint stars whose centres form an edge of $\mathcal {H}$ . A star formed by edges $A_1, \dotsc , A_{r-1}$ and centred at v is rainbow if there are distinct colours such that, for each j, all vertices of $A_j \setminus \{v\}$ are coloured $i_j$ . A constellation is rainbow if its s constituent stars are rainbow and have the same colour pattern (the set $\{i_1, \dotsc , i_{r-1}\}$ ). The conjunction of the following five assumptions implies that non-r-colourability has a sharp threshold in $\mathcal {H}_p$ :

1. (A1) Symmetry. The hypergraph $\mathcal {H}$ is symmetric in the sense that its group of automorphisms $\mathrm {Aut}(\mathcal {H})$ acts transitively on the vertex set of $\mathcal {H}$ .

2. (A2) Nonclusteredness. The hypergraph $\mathcal {H}$ is nonclustered, which means (roughly speaking) that $\mathcal {H}$ satisfies the assumptions of the hypergraph container lemma with density parameter $p_{\mathcal {H}}$ . (See Section 2.2.)

3. (A3) Weak threshold. The function $p_{\mathcal {H}}$ is a threshold for the property that $\mathcal {H}_p$ is not r-colourable.

4. (A4) Choosability of typical bounded-sized subsets. The random set $V(\mathcal {H})_{p_{\mathcal {H}}}$ a.a.s. does not contain any set W with $O(1)$ vertices for which $\mathcal {H}[W]$ is not choosable from $2$ -element lists of colours in . (See Section 2.3.)

5. (A5) The rainbow star-constellation property. Every partial r-colouring of the vertices of $\mathcal {H}$ that makes a constant proportion of its stars rainbow must make a constant proportion of its constellations rainbow as well. (See Section 2.4.)

Theorem 2.1. Let $s \geqslant 3$ and $r \geqslant 2$ be integers and let $\mathcal {H}$ be a sequence of s-uniform hypergraphs. If $\mathcal {H}$ satisfies assumptions (A1)–(A5), then there exists a function $\hat {p} = \Theta (p_{\mathcal {H}})$ such that the following holds for every positive $\varepsilon $ :

$$\begin{align*}{\mathbb{P}}\left(\mathcal{H}_p \text{ is } r\text{-colourable}\right) \to \begin{cases} 1 & \text{if } p \leqslant (1-\varepsilon)\hat{p}, \\ 0 & \text{if } p \geqslant (1+\varepsilon)\hat{p}. \end{cases} \end{align*}$$

Verifying assumptions (A1) and (A2) for our applications of the theorem will be completely straightforward. Assumption (A3) is not at all easy to check, but, for the three applications of the main theorem we consider in this work, it had been established by earlier works. Moreover, it is now standard to derive the $1$ -statement in (A3) from assumption (A2) and a property we term robust noncolourability, see Section 2.5. Verifying assumption (A4), which is closely related to establishing the $0$ -statement in (A3), takes the most effort. Assumption (A5) holds trivially when every set of $\Omega (v(\mathcal {H}))$ vertices induces $\Omega (e(\mathcal {H}))$ edges, which is the case in the context of van der Waerden’s theorem and Ramsey’s theorem for bipartite graphs. In the two remaining applications of the theorem discussed here – Schur’s theorem and Ramsey’s theorem for nonbipartite, collapsible graphs – establishing this assumption requires a nontrivial argument. Finally, let us mention that there are natural sequences of hypergraphs, for which one would expect non-r-colourability to have a sharp threshold, that satisfy assumptions (A1)–(A4), but fail to satisfy (A5), see Appendix C.

2.2 Nonclusteredness

Given a hypergraph $\mathcal {H}$ and a set $T \subseteq V(\mathcal {H})$ , we will denote by $\deg _{\mathcal {H}}(T)$ the degree of T in $\mathcal {H}$ , that is,

Further, for an integer $t \geqslant 1$ , we let $\Delta _t(\mathcal {H})$ be the maximum degree of a t-element set of vertices, defined by

We are now ready to define the notion of nonclusteredness from assumption (A2).

Definition 2.2. A sequence of nonempty, s-uniform hypergraphs $\mathcal {H}$ is called nonclustered if

$$\begin{align*}\Delta_1(\mathcal{H}) = O\left(\frac{e(\mathcal{H})}{v(\mathcal{H})}\right) \qquad \text{and} \qquad \Delta_t(\mathcal{H}) \ll p_{\mathcal{H}}^{t-1} \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})} \quad \text{for } t \in \{2, \dotsc, s-1\}. \end{align*}$$

2.3 Choosability of typical bounded-sized subsets

Recall that a hypergraph $\mathcal {G}$ is 2-choosable from a set C of colours if, for every assignment $L \colon V(\mathcal {G}) \to \binom {C}{2}$ of $2$ -element lists of colours to the vertices of $\mathcal {G}$ , there exists a proper colouring of $\mathcal {G}$ that assigns to each vertex $v \in V(\mathcal {G})$ a colour from its list $L_v$ . Given a hypergraph $\mathcal {H}$ and integers $k \geqslant 1$ and $r \geqslant 2$ , define

The precise statement of assumption (A4) is that, for every $k \geqslant 1$ ,

(1) $$ \begin{align} {\mathbb{P}}\big(V(\mathcal{H})_{p_{\mathcal{H}}} \supseteq W \text{ for some } W \in \mathcal{N}_k(\mathcal{H})\big) \to 0. \end{align} $$

In fact, our argument may require that (1) holds also when we replace $p_{\mathcal {H}}$ with some $p = \Theta (p_{\mathcal {H}})$ . Fortunately, these two statements are completely equivalent, see Lemma 4.6.

Finally, it is worth pointing out that the assumption on choosability of typical bounded-sized subsets is necessary for non- $2$ -colourability of $\mathcal {H}_p$ to have a sharp threshold at some $\hat {p} = \Theta (p_{\mathcal {H}})$ . Indeed, if (1) fails for some constant k, then the probability that $V(\mathcal {H})_p$ contains some $W \in \mathcal {N}_k(\mathcal {H})$ , which clearly makes $\mathcal {H}_p$ not $2$ -colourable, is bounded away from zero for every $p = \Omega (p_{\mathcal {H}})$ .

2.4 The rainbow star-constellation property

Our final assumption (A5) has a much less obvious connection to the problem at hand, but it conveniently fits into our framework.

A fairly straightforward calculation (Lemma 6.3) shows that every nonclustered sequence of s-uniform hypergraphs $\mathcal {H}$ contains $\Theta \big (e(\mathcal {H})^{r-1} / v(\mathcal {H})^{r-2}\big )$ many $(r-1)$ -stars and $\Theta \big (e(\mathcal {H})^{s(r-1)+1} / v(\mathcal {H})^{s(r-1)}\big )$ constellations of $(r-1)$ -stars. We will say that such a sequence $\mathcal {H}$ has the rainbow star-constellation property for r colours if every partial r-colouring of the vertices of $\mathcal {H}$ that makes a constant proportion of all its $(r-1)$ -stars rainbow also makes a constant proportion of all its constellations rainbow.

Figure 1

Stars and constellations.

2.5 The weak threshold assumption

We conclude this section with a short discussion on how assumption (A3) might possibly be derived from (A2) and (A4) and yet another ‘supersaturation’ assumption on $\mathcal {H}$ that we term robust non-r-colourability.

Definition 2.9. Given an integer $r \geqslant 2$ , we say that a sequence of hypergraphs $\mathcal {H}$ is robustly non-r-colourable if every r-colouring of the vertices of $\mathcal {H}$ makes a constant proportion of the edges of $\mathcal {H}$ monochromatic, that is, if every satisfies

$$\begin{align*}\sum_{i=1}^r e\big(\mathcal{H}[c^{-1}(i)]\big) = \Omega\big(e(\mathcal{H})\big). \end{align*}$$

The following facts can be derived from the corresponding Ramsey statements using simple averaging arguments and are thus considered folklore.

If a sequence of hypergraph is nonclustered and robustly non-r-colourable, then $O(p_{\mathcal {H}})$ is an upper bound on any threshold function of non-r-colourability. This fact can be shown by a straightforward adaptation of the argument of Nenadov and Steger [Reference Nenadov and Steger21], who showed that robust $(r+1)$ -colourability of the sequence $\mathcal {H}$ of hypergraphs representing copies of a given graph H in $K_n$ implies non-r-colourability of a typical $\mathcal {H}_p$ for all $p \gg n^{-1/m_2(H)}$ . (See also [Reference Balogh and Samotij3, Section 8] for a slightly different version of this argument that shows the exact statement of the proposition below.) For the sake of completeness, we recreate this argument in Appendix A.

Proposition 2.11. Let $r \geqslant 2$ and $s \geqslant 2$ be integers. For every nonclustered, robustly non-r-colourable sequence $\mathcal {H}$ of s-uniform hypergraphs, there exists a constant C such that, for every $p \geqslant Cp_{\mathcal {H}}$ ,

$$\begin{align*}{\mathbb{P}}\big(\mathcal{H}_p \text{ is } r\text{-colourable}\big) \leqslant \exp\left(-\Omega\big(p \cdot v(\mathcal{H})\big)\right). \end{align*}$$

It would be worth looking into the following problem, motivated by [Reference Nenadov and Steger21, Lemma 6] and [Reference Nenadov, Person, Škorić and Steger20, Meta-Theorem].

One could consider strengthening the assumption of being nonclustered by further assuming that, for each $t \in \{2, \dotsc , s-1\}$ , the inequality $\Delta _t(\mathcal {H}) \ll p_{\mathcal {H}}^{t-1} \cdot \frac {e(\mathcal {H})}{v(\mathcal {H})}$ hides some polynomial (in $v(\mathcal {H})$ ) factor. We remark that the three families of sequences of hypergraphs from the statement of Fact 2.4 all enjoy such strengthened nonclusteredness property. It is not unlikely that one can extend the methods from subsection 7.4 and from the subsequent works [Reference Kuperwasser and Samotij17, Reference Kuperwasser, Samotij and Wigderson18] to prove the statement under this strengthened assumption.

3.1 Boosters and dichotomy

Our point of departure will be Friedgut’s criterion, in Bourgain’s formulation, which tells us that there is a positive constant c, a sequence p satisfying

(2) $$ \begin{align} c \leqslant {\mathbb{P}}\left(V_p \text{ is not } r\text{-colourable}\right) \leqslant 1- c, \end{align} $$

and a family $\mathbb {B}$ of constant-sized subsets of V such that

(3) $$ \begin{align} {\mathbb{P}} \left( \exists B \in \mathbb{B} \text{ s.t.\ } B \subseteq V_p\right)> c \end{align} $$

and every $B \in \mathbb {B}$ is a booster.

Definition 3.1. Given $\delta> 0$ and $p \in [0,1]$ , a set $B \subseteq V$ is called a $(p,\delta )$ -booster if

$$\begin{align*}{\mathbb{P}} \left(V_p \text{ is not } r\text{-colourable} \mid B \subseteq V_p \right)> {\mathbb{P}}\left(V_p \text{ is not } r\text{-colourable} \right) + \delta. \end{align*}$$

Observe that assumption (A3) and (2) imply that $p = \Theta (p_{\mathcal {H}})$ . Using the symmetry assumption (A1), we will expand on Friedgut’s criterion and show that a typical sample $Z \sim V_p$ exemplifies a sort of dichotomy, in the following precise sense.

Step I. There are constants $\alpha , \varepsilon> 0$ , an integer K, and $p = \Theta (p_{\mathcal {H}})$ such that, for any family $\mathcal {F} \subseteq \binom {\mathcal {H}}{\leqslant K}$ with

(4) $$ \begin{align} {\mathbb{P}} \left( \exists B \in \mathcal{F} \text{ s.t.\ } B\subseteq V_p \right) < \alpha, \end{align} $$

there is a set $B_0 \in \binom {V(\mathcal {H})}{\leqslant K} \setminus \mathcal {F}$ such that the following holds. For infinitely many values N, the set $Z \sim V_p$ satisfies the following with probability larger than $\alpha $ :

$$\begin{align*}{\mathbb{P}} \left(Z \cup h(B_0) \text{ is not } r\text{-colourable} \mid Z \right)> \alpha, \end{align*}$$

where h is taken u.a.r. from the set of symmetries of $\mathcal {H}$ , and

$$\begin{align*}{\mathbb{P}} \left(Z \cup V_{\varepsilon p} \text{ is not } r\text{-colourable} \mid Z\right) \leqslant 1/2. \end{align*}$$

In other words, we are guaranteed the existence of two sets, $B_0$ and Z, with two properties that seem at odds. While a positive proportion of the symmetric copies of the constant-sized $B_0$ interact with Z – that is, Z ceases to be r-colourable once we add them – the probability that the random set $V_{\varepsilon p}$ interacts with Z is bounded away from one. We will call these interacting symmetric copies of $B_0$ activated boosters.

Furthermore, we are allowed to trim some undesirable properties from both $B_0$ and Z. In the case of $B_0$ , this can be done by requiring that $B_0 \notin \mathcal {F}$ whereas in the case of Z, this can be done as $Z \sim V_p$ satisfies the assertion with probability bounded away from zero. We should remark at this point that the family $\mathcal {F}$ we are going to choose will be symmetric, that is, if $B \in \mathcal {F}$ , then $h(B) \in \mathcal {F}$ for every $h \in \mathrm {Aut}(\mathcal {H})$ . Therefore, if $B_0 \notin \mathcal {F}$ , then the same is true for every other symmetric copy of it.

Our aim is to use these two statements to get a contradiction. Specifically, we will show that, with a suitable choice of properties for $B_0$ and Z that exploit assumptions (A2)–(A5), the existence of many activated boosters implies that $Z \cup V_{\varepsilon p}$ is not r-colourable with probability arbitrarily close to one. The methodology of the argument will be very much in tune with the previous works of Friedgut, Hán, Person, and Schacht [Reference Friedgut, Hàn, Person and Schacht10], Schacht and Schulenburg [Reference Schacht and Schulenburg28], and Schulenburg [Reference Schulenburg29]. However, in order to extend these results we require some novel ideas. Adding the choosability assumption (A4) to the mix will allow us to argue for sharpness in three or more colours. At the same time, the introduction of the rainbow star-constellation property (A5) empowers the argument to target more complex combinatorial structures. As we will soon see, certain aspects of the argument simplify if the Turán problem that is associated with the structure is degenerate (as is the case with arithmetic progressions or bipartite graphs). The previous methods were able to argue about such cases, and even go further to tackle structures that are nearly degenerate (e.g., Schur triples and nearly bipartite graphs). Property (A5) encodes a weaker sense of degeneracy that generalises the previous cases and also applies for new structures, such as cliques of any size.

We will present our proof in two rounds. The first round will be a (spoiler alert) failed attempt, which will still show in essence how to utilise the assumption about choosability of typical subsets of bounded size (which we will enforce on $B_0$ via an appropriate choice of $\mathcal {F}$ ) to gain structural information on proper r-colourings of Z. The second round will address the breaking point of that approach, a very large union bound over r-colourings of Z, and remedy it using the Hypergraph Container Lemma of Saxton and Thomason [Reference Saxton and Thomason27] and also of Balogh, Morris, and Samotij [Reference Balogh, Morris and Samotij2]. This approach will, in turn, require us to strengthen one of the claims made in the first round.

3.2 First attempt

We start with Step (I) and get Z and a family of interacting boosters, all of which are symmetric copies of $B_0$ . Define the hypergraph $\mathcal {B}$ on the vertex set V whose edges are all symmetric copies of $B_0$ , that is, all $h(B_0)$ with $h \in \mathrm {Aut}(\mathcal {H})$ . Since $\mathcal {H}$ is symmetric, $\mathcal {B}$ is regular and, therefore,

$$\begin{align*}\Delta_1(\mathcal{B}) = |B_0| \cdot \frac{e(\mathcal{B})}{v(\mathcal{H})} \leqslant K\cdot N^{-1} \cdot e(\mathcal{B}). \end{align*}$$

We will impose some structural assumptions on $B_0$ . It is natural to require that $B_0$ is r-colourable, since otherwise $Z \cup h(B_0)$ would be not r-colourable with probability one, which would in turn suggest that the threshold is actually coarse. (This was the only assumption on $B_0$ imposed in previous works [Reference Friedgut, Hàn, Person and Schacht10, Reference Schacht and Schulenburg28, Reference Schulenburg29].) We will go one step further. Instead of ensuring that $B_0$ is only r-colourable, we will make use of assumption (A4) and require that it is $2$ -choosable from lists in . We may do so as (A4) implies that the family $\mathcal {F}$ comprising all non- $2$ -choosable subsets of V with at most K vertices satisfies the condition (4) in Step (I). Note that this property is symmetric, so requiring it from $B_0$ guarantees that it is fulfilled by all $B \in \mathcal {B}$ .

Let $\mathcal {B}_Z \subseteq \mathcal {B}$ comprise only the copies of $B_0$ that interact with Z, that is,

The first assertion of Step (I) translates to $e(\mathcal {B}_Z)> \alpha \cdot e(\mathcal {B})$ . One of the desirable properties of Z would allow us to find a subfamily $\mathcal {B}_Z' \subseteq \mathcal {B}_Z$ of our activated boosters – satisfying $e(\mathcal {B}_Z') \geqslant \alpha /2 \cdot e(\mathcal {B})$ – whose members interact with Z in a very well-behaved manner. First, if $B \in \mathcal {B}_Z'$ , then B and Z are disjoint. Further, suppose that some activated booster $B \in \mathcal {B}_Z$ does not intersect Z. Since both Z and B are r-colourable, the fact that $Z \cup B$ is not means that there is an edge of $\mathcal {H}[Z \cup B]$ that intersects both B and Z. Call the set of all such edges the interface between B and Z. The second desirable property, which we may impose using assumption (A2), is that, if $B \in \mathcal {B}_Z'$ , then each edge in the interface between B and Z has exactly one vertex in B (and the remaining vertices in Z).

Let $\psi \in \mathrm {Col}(Z)$ be a proper colouring of Z. We say that $\psi $ forces a vertex $v \in V$ to the colour if v is threatened by $\psi ^{-1}(j)$ for every $j \neq i$ . (Equivalently, $\psi $ forces v to the colour i if and only if v is the centre of an i-rainbow star.) The motivation behind the definition is the following fact: If $\psi $ forces v to the colour i, then every proper colouring of $Z \cup \{v\}$ that extends $\psi $ must assign the colour i to v. Let $F_i(\psi )$ denote the set of vertices that $\psi $ forced to the colour i and let be the set of forced vertices.

We prove this statement in two steps. First, we show that in order to force a constant fraction of the vertices, it is enough to force at least one vertex in a constant fraction of the activated boosters. Second, we show that every colouring $\psi \in \mathrm {Col}(Z)$ forces a vertex inside every activated booster.

Proof. Let $\beta> 0$ be some constant such that F intersects $\beta \cdot e(\mathcal {B}^{\prime }_Z)$ activated boosters. Since each vertex belongs to at most $\Delta _1(\mathcal {B})$ boosters, we have

$$\begin{align*}\beta \cdot \alpha/2 \cdot e(\mathcal{B}) \leqslant \beta \cdot e(\mathcal{B}_Z') \leqslant |F| \cdot \Delta_1(\mathcal{B}) \leqslant |F| \cdot K \cdot N^{-1} \cdot e(\mathcal{B}), \end{align*}$$

which implies that F has $\Omega (N)$ elements.

By the pigeonhole principle, one of the forced sets, say $F_i(\psi )$ , has $\Omega (N)$ vertices. We would like to show that this set induces $\Omega (e(\mathcal {H}))$ edges. (This is essentially equivalent to $\Omega (e(\mathcal {H}))$ edges being the base edge of an i-rainbow constellation.) Such a claim is always true when $\mathcal {H}$ describes a ‘degenerate’ structure; e.g., if $\mathcal {H}$ is the hypergraph of copies of a bipartite graph or when $\mathcal {H}$ is the hypergraph of arithmetic progressions. Unfortunately, some hypergraphs of interest (e.g., the hypergraph of copies of any non-bipartite graph) contain independent sets of cardinality $\Omega (N)$ . However, using the assumption that $\mathcal {H}$ has the rainbow star-constellation property, we will be able to argue that a.a.s. $Z \sim V_p$ has the property that, for every $\psi \in \mathrm {Col}(Z)$ , every large set $F_i(\psi )$ must induce many edges. With foresight, we state a stronger version of this property that extends also to partial r-colourings of Z.

The proof of this statement will appear in Section 6. We will just mention that it employs the Hypergraph Container Lemma to transfer the supersaturation statement given by the rainbow star-constellation assumption into the sparse regime.

Note now that if $e \subseteq F_i(\psi )$ is an edge, then $\psi $ cannot be extended to e while staying proper. This is because the elements of $F_i(\psi )$ must all be coloured i, but this makes e monochromatic. The fact that $F_i(\psi )$ contains $\Omega (e(\mathcal {H}))$ edges, together with the assumption that the hypergraph $\mathcal {H}$ is not clustered, makes it extremely unlikely that such an edge will fail to appear in $V_{\varepsilon p}$ . The following estimate follows from Janson’s inequality (Theorem 4.9).

Thus far, we have demonstrated that the probability that a given $\psi \in \mathrm {Col}(Z)$ could be extended to $Z \cup V_{\varepsilon p}$ is bounded from above by $\exp (-\Omega (pN))$ . However, in order to get the desired contradiction to the first assertion of Step (I), we would like to show that the probability that some proper colouring of Z can be extended to $V_{\varepsilon p}$ tends to zero. To this end, let us take the union bound over all proper colourings. Alas, the only bound we have at our disposal is $|\mathrm {Col}(Z)| = \exp (O(|Z|)) = \exp (O(pN))$ , which is not good enough. Moreover, we cannot significantly improve the upper bound established by Lemma 3.6, since the set $V_{\varepsilon p}$ is empty with probability approximately $\exp (-\varepsilon p N)$ .

3.3 Second attempt

Since the breaking point of our first attempt was the union bound over all proper colourings of Z, we will try to make this union bound more efficient by excluding many colourings in $\mathrm {Col}(Z)$ at once. To this end, note that if $\psi _0$ were a partial colouring of Z that forced $\Omega (N)$ vertices to some colour, we could still apply Step (III) and Lemma 3.6 to learn that the probability that $\psi _0$ can be extended to $V_{\varepsilon p}$ is at most $\exp (-\Omega (pN))$ . Moreover, if $\psi _0$ cannot be extended to $V_{\varepsilon p}$ , then neither can any proper colouring $\psi \in \mathrm {Col}(Z)$ that extends $\psi _0$ . It thus suffices to find a family of $\exp (o(|Z|)) = \exp (o(pN))$ partial colourings of Z, each forcing a constant proportion of the vertices to some colour, such that every element of $\mathrm {Col}(Z)$ is an extension of some member of the family.

As was hinted before, we will employ the Container Lemma [Reference Balogh, Morris and Samotij2, Reference Saxton and Thomason27] to find such a family. To this end, we will identify every colouring with the set $\{ (z, \psi (z)) \colon z \in Z\}$ and define a hypergraph $\mathcal {T}$ on the vertex set such that every $\psi \in \mathrm {Col}(Z)$ will correspond to an independent set of $\mathcal {T}$ . The Container Lemma will allow us to construct a family $\mathcal {C}$ of subsets of , which we will call containers, such that:

1. (C1) Any independent set of $\mathcal {T}$ , and thus every $\psi \in \mathrm {Col}(Z)$ , is contained in some $C \in \mathcal {C}$ .

2. (C2) Every container $C \in \mathcal {C}$ induces only a small fraction of the edges of $\mathcal {T}$ .

We will say that a set is a restricted colouring if every $z \in Z$ has at least one ‘available’ colour in C, that is, if $(z, i) \in C$ for some . Note that a set that is not a restricted colouring cannot contain any colouring of Z. Given a restricted colouring , we define its determined colouring $\psi _C$ to be the maximal partial colouring agreed upon by all the colourings contained by C. (That is, $\psi _C(z) = i$ if and only if i is the unique colour such that $(z, i) \in C$ .) Note that (C1) implies that every $\psi \in \mathrm {Col}(Z)$ extends the determined colouring $\psi _C$ of some $C \in \mathcal {C}$ . The crux of our argument is showing that one may define $\mathcal {T}$ in such a way that condition (C2) implies that, for every $C \in \mathcal {C}$ , the determined colouring $\psi _C$ forces many vertices to some colour, so that we can apply Step (III).

The precise definition of the hypergraph $\mathcal {T}$ is somewhat technical, but the idea behind it is fairly straightforward. Given a booster $B \in \mathcal {B}_Z'$ , we will say that colourings and are consistent if no edge in the interface between B and Z is monochromatic under $\psi \cup \phi $ . Since each such interface edge has exactly one vertex in B (by the definition of $\mathcal {B}_Z'$ ), two colourings $\psi $ and $\phi $ are consistent if and only if no vertex $b \in B$ is threatened by $\psi ^{-1}(\phi (b))$ . A key observation is that the fact that two colourings $\psi $ and $\phi $ as above are consistent always has a small ‘witness’ set in . Indeed, for a given $\phi $ , one can certify that $\psi $ is consistent with $\phi $ by specifying the values of $\psi $ on vertices of Z that belong to edges of the interface between B and Z. Such minimal ‘witness’ sets for all $B \in \mathcal {B}_Z'$ and all proper colourings $\phi \in \mathrm {Col}(B)$ are the edges of our hypergraph $\mathcal {T}$ . The fact that every $B \in \mathcal {B}_Z'$ is an activated booster means that no two colourings $\psi \in \mathrm {Col}(Z)$ and $\phi \in \mathrm {Col}(B)$ are consistent and, consequently, every $\psi \in \mathrm {Col}(Z)$ is an independent set of $\mathcal {T}$ .

We say that a restricted colouring C inter-activates a booster $B \in \mathcal {B}_Z'$ if every proper colouring $\phi \in \mathrm {Col}(B)$ is inconsistent with each colouring of Z contained in C. Our definition of $\mathcal {T}$ guarantees that a restricted colouring C will induce an edge in $\mathcal {T}$ for every booster $B \in \mathcal {B}_Z'$ which it fails to inter-activate. (This edge will be a witness to some pair $\phi \in \mathrm {Col}(B)$ and $\psi \subseteq C$ being consistent.) Since every container $C \in \mathcal {C}$ induces only a small proportion of all edges of $\mathcal {T}$ , it must therefore inter-activate all but a small fraction of all boosters in $\mathcal {B}_Z'$ . Finally, an argument similar to the one used in the proof of Lemma 3.5 shows that if C inter-activates a booster B, then $\psi _C$ must force at least one vertex in B to some colour. (A key insight here is realising that the property that C inter-activates a booster depends only on the determined colouring $\psi _C$ .) This implies, by Lemma 3.4, that $\psi _C$ must force $\Omega (N)$ vertices.

Our final concern is that the family $\mathcal {C}$ of containers will have at most $\exp (o(|Z|))$ elements. In order to guarantee this, we will need to demonstrate some control over the edge set of $\mathcal {T}$ . In fact, it will be sufficient to bound the largest size of an edge of $\mathcal {T}$ by an absolute constant and to show that $\Delta _1(\mathcal {T}) \cdot |Z| = O(e(\mathcal {T}))$ and $\Delta _2(\mathcal {T}) \cdot |Z| \ll e(\mathcal {T})$ . Luckily, this will be possible yet again by trimming further undesirable properties from Z in Step (I). For example, the expected number of interacting edges between $V_p$ and B is bounded by some constant and consequently the witness sets will also be constant sized. Using this and other related properties of Z and the booster family, we will be able to show that the hypergraph $\mathcal {T}$ is ‘well-behaved’, which will allow us to use the Container Lemma to derive the following.

Our union bound argument is now brought back to life and the proof is finally settled.

3.4 Finalising the argument

Our goal for the next three sections is to tie up the loose ends of the proof, by filling in the gaps in the outline presented above. Recall that we want to show that, assuming non-r-colourability does not have a sharp threshold in $\mathcal {H}_p$ , the following statements hold for $Z \sim V_p$ with probability bounded away from zero:

1. (I) For some constant $\varepsilon> 0$ , the hypergraph $\mathcal {H}\big [Z \cup V_{\varepsilon p}\big ]$ is r-colourable with probability at least $1/2$ .

2. (II) Each proper r-colouring of Z extends one of $\exp (o(|Z|))$ partial colourings, each of which forces $\Omega (N)$ elements in V to some colour.

3. (III) Each of our partial colourings admits a set of $\Omega (e(\mathcal {H}))$ edges whose all vertices are forced to the same colour.

As we argued above, this would lead to a contradiction. Indeed, each individual partial colouring from the family cannot be extended to any set that contains one of the $\Omega (e(\mathcal {H}))$ ‘forced’ edges. By Lemma 3.6, the probability that $V_{\varepsilon p}$ omits all these edges is at most $\exp \big (-\Omega (pN)\big )$ . A union bound over all partial colourings yields that $\mathcal {H}\big [Z \cup V_{\varepsilon p}\big ]$ is not r-colourable with probability tending to one.

We prove statements (I), (II), and (III) over the course of Sections 4, 5, and 6. More precisely, Section 4 contains the formal statements of Friedgut’s sharp threshold criterion, along with a routine corollary thereof (Step (I) in the proof outline), as well as the statements of the Hypergraph Container Lemma and Janson’s inequality, of which Lemma 3.6 is a simple corollary.

Next, in Section 5, we show how the abundance of boosters can be used to construct the efficient family of partial colourings (the refined version of Step (II) in the proof outline). In order to formally phrase this result using the notions from assumption (A5), we shift the terminology from forced vertices to (centres of) rainbow stars. The result is the following theorem.

Finally, Section 6 provides a proof of the following sparse analogue of the rainbow star-constellation property (Step (III) in the proof outline). We remark again that this theorem is trivial if the hypergraph $\mathcal {H}$ is ‘degenerate’ in the sense that every subset of $\Omega (N)$ vertices induces $\Omega (e(\mathcal {H}))$ edges. As a result, we are spared from a bulk of the proof when $\mathcal {H}$ is the hypergraph of copies of a bipartite graph or the hypergraph of arithmetic progressions of a prescribed length.

Theorems 3.7 and 3.8, supplemented with Lemma 3.6, imply Theorem 2.1.

4.1 Coarse thresholds

The first piece of machinery that we require is a characterisation of properties with coarse thresholds. In the proof outline, this was captured by Step (I), which we aim to formalise and prove here. In general, Friedgut’s work [Reference Friedgut8] and Bourgain’s subsequent extension of it [Reference Friedgut8, Appendix] provide a criterion for the appearance of a sharp threshold. The criterion states that every property that fails to have a sharp threshold must correlate with a ‘local’ property, that is, the appearance of a bounded-sized subset. In other words, if a property $\mathcal {P}$ has a coarse threshold, then there is another, local, property $\mathcal {P}'$ – with the same threshold as $\mathcal {P}$ – such that both are positively correlated.

For our purpose, we use a reformulation of Bourgain’s aforementioned result, which appears in [Reference Friedgut9]. We will introduce a relevant definition and then state the theorem.

Definition 4.1 (Boosters).

Suppose that $\mathcal {P}$ is a property of subsets of a finite set V. Given a $p \in [0,1]$ and a positive number $\delta $ , we call a set $B \subseteq V$ a $(p,\delta )$ -booster if

$$\begin{align*}{\mathbb{P}}(V_p \cup B \in \mathcal{P}) \geqslant {\mathbb{P}}(V_p \in \mathcal{P}) + \delta. \end{align*}$$

Theorem 4.2 (The Sharp Threshold Criterion).

For all positive $\alpha $ and C, there exist positive $\delta $ , $\eta $ , $p_0$ , and K such that the following holds. Suppose that $\mathcal {P}$ is a monotone property of subsets of a finite set V and, for each $p \in [0,1]$ , let . If, for some $0< p \leqslant p_0$

$$\begin{align*}\alpha \leqslant \mu(p) \leqslant 1-\alpha \qquad \text{and} \qquad \mu'(p) \leqslant C/p, \end{align*}$$

then there is a family $\mathcal {B} \subseteq \binom {V}{\leqslant K}$ satisfying

$$\begin{align*}{\mathbb{P}}(B \subseteq V_p \text{ for some } B \in \mathcal{B})> \eta \end{align*}$$

such that every $B \in \mathcal {B}$ is a $(p,\delta )$ -booster for $\mathcal {P}$ .

We will now use the criterion to derive a general result in the spirit of Step (I) of the proof outline. First, it guarantees the existence of boosters with ‘typical’ characteristics. Second, it asserts the following dichotomy for properties $\mathcal {P}$ with coarse thresholds: while there are many (constant-sized) boosters whose addition to $V_p$ lands us immediately in $\mathcal {P}$ , adding to $V_p$ a random set of density $\varepsilon p$ does not increase the probability of being in $\mathcal {P}$ substantially. We again introduce a definition and move on to stating and proving the result.

Proof. Suppose that $\mathcal {P}$ does not have a sharp threshold. This means that there are constants $c_1 < c_2$ and $\alpha> 0$ such that

$$\begin{align*}\alpha \leqslant \mu(c_1 \cdot \hat{p}) \leqslant \mu(c_2 \cdot \hat{p}) \leqslant 1-\alpha \end{align*}$$

on some infinite subsequence. Let .

Claim 4.5. For every $\gamma> 0$ , there is a $p \in (c_1 \cdot \hat {p}, c_2 \cdot \hat {p})$ such that

1. (a) $\mu '(p) \leqslant C/p$ and

2. (b) $\mu (p+\gamma p) - \mu (p) \leqslant C \cdot \gamma $

Proof. Fix an arbitrary $\gamma '$ satisfying $0 < \gamma ' < (c_2-c_1)/2$ . Since $0 \leqslant \mu \leqslant 1$ , we have

$$\begin{align*}\int_{c_1}^{c_2-\gamma'} \big(\mu((x+\gamma') \cdot \hat{p}) - \mu(x \cdot \hat{p})\big)\, dx \leqslant \int_{c_2-\gamma'}^{c_2} \mu(x \cdot \hat{p}) \, dx \leqslant \gamma'. \end{align*}$$

In particular, there must be a $p'$ satisfying $c_1 \cdot \hat {p} < p' < (c_2-\gamma ') \cdot \hat {p}$ such that

$$\begin{align*}\mu(p' + \gamma' \cdot \hat{p}) - \mu(p') \leqslant \frac{\gamma'}{c_2-c_1-\gamma'} \leqslant \frac{2\gamma'}{c_2-c_1}. \end{align*}$$

Further, as $\mu $ is increasing, there must be a $p \in (p', p' + (\gamma '/2) \cdot \hat {p}) \subseteq (c_1 \cdot \hat {p}, c_2 \cdot \hat {p})$ with

$$\begin{align*}\mu'(p) \leqslant \frac{\mu(p'+(\gamma'/2)\cdot\hat{p})-\mu(p')}{(\gamma'/2) \cdot \hat{p}} \leqslant \frac{4}{(c_2-c_1) \cdot \hat{p}} \leqslant \frac{4c_2}{(c_2-c_1) \cdot p} = \frac{C}{p}, \end{align*}$$

as required. Finally, we show that the value p chosen above also satisfies inequality (b). If $\gamma \geqslant 1/C$ , then the inequality is vacuous. Otherwise, if $\gamma < 1/C$ , we let $\gamma ' = 2c_2\gamma $ . Since $\gamma ' < (c_2-c_1)/2$ , by the definition of C, we have

$$\begin{align*}p + \gamma p \leqslant p + \gamma \cdot c_2 \hat{p} = p + (\gamma'/2) \cdot \hat{p} \leqslant p' + \gamma' \cdot \hat{p}. \end{align*}$$

Since $p> p'$ and $\mu $ is increasing, we have

$$\begin{align*}\mu(p+\gamma p) - \mu(p) \leqslant \mu(p' + \gamma' \cdot \hat{p}) - \mu(p') \leqslant \frac{2\gamma'}{c_2-c_1} = C\gamma, \end{align*}$$

as desired.

Let $\mathcal {B}$ be an arbitrary nonempty subset of the family of $(p, \delta )$ -boosters supplied by Theorem 4.2. Fix a small positive constant $\gamma $ , let B be a uniformly chosen random element of $\mathcal {B}$ , and, for each $Z \subseteq V$ , define

$$\begin{align*}f(Z) = {\mathbb{P}}(Z \cup B \in \mathcal{P}) \qquad \text{and} \qquad g(Z) = {\mathbb{P}}(Z \cup V_{\gamma p} \in \mathcal{P}). \end{align*}$$

Our goal is to show that, for some positive constant $\varepsilon $ ,

$$\begin{align*}{\mathbb{P}}\big(f(V_p) \geqslant \varepsilon \text{ and } g(V_p) \leqslant 1/2\big) \geqslant \varepsilon. \end{align*}$$

To this end, note first that Markov’s inequality and the definition of a $(p, \delta )$ -booster imply

$$\begin{align*}{\mathbb{P}}\big(f(V_p) < \varepsilon\big) = {\mathbb{P}}\big(1 - f(V_p)> 1 - \varepsilon\big) \leqslant \frac{1-\mathbb{E}[f(V_p)]}{1-\varepsilon} \leqslant \frac{1-\mu(p)-\delta}{1-\varepsilon}. \end{align*}$$

Since $g(Z) = 1$ when $Z \in \mathcal {P}$ and since $V_p \cup V_{\gamma p}$ is stochastically dominated by $V_{p+\gamma p}$ , we have, again by Markov’s inequality,

Finally,

$$\begin{align*}\begin{aligned} {\mathbb{P}}\big(f(V_p) \geqslant \varepsilon \text{ and } g(V_p) \leqslant 1/2\big) & \geqslant 1 - {\mathbb{P}}\big(f(V_p) < \varepsilon\big) - {\mathbb{P}}\big(g(V_p)> 1/2\big) \\ & \geqslant 1 - \frac{1-\mu(p)-\delta}{1-\varepsilon} - \mu(p) - 2C\gamma \\ & \geqslant \delta - \frac{\varepsilon}{1-\varepsilon} -2C\gamma \geqslant \delta/2 \geqslant \varepsilon, \end{aligned} \end{align*}$$

where the last two inequalities hold provided that $\varepsilon $ and $\gamma $ are sufficiently small (as functions of $\delta $ and C only).

Finally, we will make use of the following simple lemma, which formalises the fact that all ‘local’ properties (i.e., properties of containing a bounded-sized subset from a given family) have coarse thresholds.

Proof. We couple $V_p$ and $V_{cp}$ by viewing $V_{cp}$ as the random subset $(V_p)_c \subseteq V_p$ . Therefore,

$$\begin{align*}\frac{\mu(cp)}{\mu(p)} = {\mathbb{P}}(\exists B \in \mathcal{F} \colon B \subseteq (V_p)_c \mid \exists B \in \mathcal{F} \colon B \subseteq V_p) \geqslant c^K.\\[-42pt] \end{align*}$$

4.2 Containers

Our second main tool is a hypergraph container lemma for almost independent sets. The standard versions of the container lemma, proved independently by Balogh, Morris, and Samotij [Reference Balogh, Morris and Samotij2] and by Saxton and Thomason [Reference Saxton and Thomason27], assert that every uniform hypergraph $\mathcal {G}$ admits a relatively small collection of containers for independent sets, that is, a family $\mathcal {C}$ of subsets of $V(\mathcal {G})$ , each containing only few edges of $\mathcal {G}$ , such that every independent set of $\mathcal {G}$ is contained in some member of $\mathcal {C}$ . Here, we will require a strengthening of this result that supplies a small collection of containers for the larger family of almost independent sets. Such stronger version of the container lemma was proved by Saxton and Thomason. Since the precise phrasing of this result that best fits our framework, Theorem B below, differs somewhat from [Reference Saxton and Thomason27, Corollary 3.6], we include a short derivation of the former from the latter in Appendix B.

Theorem 4.7. For every positive integer k and all positive reals $\varepsilon $ and K, there exist an integer t and a positive real $\delta $ such that the following holds. Suppose that a nonempty k-uniform (multi)hypergraph $\mathcal {G}$ with vertex set V and a positive real $\tau $ satisfy

$$\begin{align*}\Delta_\ell(\mathcal{G}) \leqslant K \tau^{\ell-1} \cdot \frac{e(\mathcal{G})}{v(\mathcal{G})} \end{align*}$$

for every . Then, there exists a function $f \colon \mathcal {P}(V)^t \to \mathcal {P}(V)$ with the following properties:

1. (i) For every set $I \subseteq V$ satisfying $e(\mathcal {G}[I]) \leqslant \delta \tau ^k e(\mathcal {G})$ , there are $S_1, \dotsc , S_t \subseteq I$ with at most $\tau v(\mathcal {G})$ elements each such that $I \subseteq f(S_1, \dotsc , S_t)$ .

2. (ii) For every $S_1, \dotsc , S_t \subseteq V$ , the set $f(S_1, \dotsc , S_t)$ induces fewer than $\varepsilon e(\mathcal {G})$ edges in $\mathcal {G}$ .

4.3 Janson’s inequality

The final auxiliary result required by our argument is the well-known concentration inequality of Janson, which gives strong upper bounds on the lower tail probabilities of random variables counting how many sets from a given collection are contained in a binomial random subset. The version of Janson’s inequality stated below differs from [Reference Janson16, Theorem 1] in the choice of notation, but it is otherwise equivalent.

Definition 4.8. Suppose that $\mathbf {A} = (A_1, \dotsc , A_k)$ is a sequence of (not necessarily distinct) events. The pseudo-variance of $\mathbf {A}$ is

where the sum ranges over all ordered pairs such that $A_i$ and $A_j$ are not independent.

Theorem 4.9 (Janson’s inequality [Reference Janson16]).

Suppose that $\Omega $ is a finite set, let $B_1, \dotsc , B_k$ be a sequence of (not necessarily distinct) subsets of $\Omega $ , and let $R \sim \Omega _p$ for some $p \in (0,1)$ . For each $i \in [k]$ , let $X_i$ be the indicator of the event $A_i$ that $B_i \subseteq R$ and let . Then, for any $0 \leqslant t \leqslant \mathbb {E}[X]$ ,

$$\begin{align*}{\mathbb{P}}\big( X\leqslant \mathbb{E}[X]-t \big) \leqslant \exp \left(-\frac{t^2}{2\mathrm{Var}'(A_1, \dotsc, A_k)}\right). \end{align*}$$

We finish this section by deriving the following generalisation of Lemma 3.6.

Lemma 4.10. Suppose that $s \geqslant 2$ and let $\mathcal {H}$ be a sequence of s-uniform hypergraphs that satisfies assumption (A2). If $p = \Theta (p_{\mathcal {H}})$ , then, for every $\mathcal {H}' \subseteq \mathcal {H}$ with $\Omega (e(\mathcal {H}))$ edges,

$$\begin{align*}{\mathbb{P}}\big(e(\mathcal{H}^{\prime}_p) = 0\big) = \exp\left(-\Omega\big(p \cdot v(\mathcal{H})\big)\right). \end{align*}$$

Proof. Write V for $V(\mathcal {H})$ and X for the number of edges of $\mathcal {H}^{\prime }_p = \mathcal {H}'[V_p]$ . We bound the expectation of X from below as follows:

Further, we bound the pseudo-variance of the sequence $\mathbf {A}$ of events $B \subseteq V_p$ for all edges $B \in \mathcal {H}'$ from above:

$$\begin{align*}\mathrm{Var}'(\mathbf{A}) = \sum_{i= 1}^s \sum_{\substack{B, B' \in \mathcal{H}' \\ |B \cap B'| = i}} {\mathbb{P}}(B \cup B' \subseteq V_p) = O\left( \sum_{i=1}^s p^{2s-i} \cdot e(\mathcal{H}) \cdot \Delta_i(\mathcal{H}) \right). \end{align*}$$

Since $p = \Theta (p_{\mathcal {H}})$ and $\mathcal {H}$ is nonclustered,

Thus, Janson’s inequality gives us that ${\mathbb {P}}(X = 0) = \exp \left (- \Omega \big (p \cdot v(\mathcal {H})\big ) \right )$ .

5.1 Setup

Suppose that s, r, and $\mathcal {H}$ are as in the statement of the theorem. For the sake of brevity, throughout this section, we shall write V in place of $V(\mathcal {H})$ . Assume that non-r-colourability does not have a sharp threshold in $\mathcal {H}_p$ . Since $p_{\mathcal {H}}$ is a threshold function for this property, by assumption (A3), Proposition 4.4 supplies constants $\delta $ , $\varepsilon $ , and K and an infinite sequence $p = \Theta (p_{\mathcal {H}})$ satisfying (1) and (2) in the proposition. Since $\mathcal {H}$ satisfies assumption (A4), invoking (1) in Proposition 4.4 with $\mathcal {F}$ being the family $\mathcal {N}_K(\mathcal {H})$ defined in Section 2.3, we obtain a $(p,\delta )$ -booster $B_0 \subseteq V$ of cardinality at most K such that $\mathcal {H}[B_0]$ is $2$ -choosable from .

Let $\mathcal {B}$ be the (multi)hypergraph on V whose edges are the images of $B_0$ via all automorphisms of $\mathcal {H}$ . Since non-r-colourability is preserved under automorphisms of $\mathcal {H}$ , every edge of $\mathcal {B}$ is also a $(p, \delta )$ -booster. Moreover, since $\mathcal {H}$ is symmetric, the hypergraph $\mathcal {B}$ is degree-regular and, consequently,

$$\begin{align*}\Delta_1(\mathcal{B}) = |B_0| \cdot \frac{e(\mathcal{B})}{v(\mathcal{H})} \leqslant K \cdot \frac{e(\mathcal{B})}{v(\mathcal{H})}. \end{align*}$$

Let $\mathcal {Z}$ be the family of all subsets $Z \subseteq V$ such that $\mathcal {H}[Z \cup V_{\varepsilon p}]$ is r-colourable with probability at least $1/2$ , but, for at least $\varepsilon $ -proportion of $B \in \mathcal {B}$ , the hypergraph $\mathcal {H}[Z \cup B]$ is not r-colourable. Property (2) in Proposition 4.4 states that ${\mathbb {P}}(V_p \in \mathcal {Z}) \geqslant \varepsilon $ .

Finally, we define several constants. Let $K_{\mathcal {H}}$ be a constant satisfying

$$\begin{align*}\Delta_1(\mathcal{H}) \leqslant K_{\mathcal{H}} \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})} \qquad \text{and} \qquad \frac{p}{p_{\mathcal{H}}} \leqslant K_{\mathcal{H}}. \end{align*}$$

Further, pick $\lambda = \varepsilon ^2/8$ and let L be an integer satisfying

(5) $$ \begin{align} \frac{\big(K \cdot K_{\mathcal{H}}^s\big)^L}{L!} \leqslant \lambda. \end{align} $$

5.2 A recap of the proof outline

Our goal now is to utilize the boosters in $\mathcal {B}$ in order to construct, for a typical $Z \in \mathcal {Z}$ , a family $\Psi $ of $\exp \big (o(|Z|)\big )$ partial colourings of Z that satisfies (a) and (b) from the statement of the theorem.

Given a $Z \in \mathcal {Z}$ , let

be the family of boosters from $\mathcal {B}$ that are active for Z. By definition, no proper -colouring of Z can be extended to $Z \cup B$ , for any $B \in \mathcal {B}_Z$ . The assumption that $\mathcal {H}[B]$ is $2$ -choosable from implies that, if a proper partial -colouring $\psi _0$ of Z cannot be extended to a booster in $B \in \mathcal {B}_Z$ , then at least one vertex of B must be the centre of a rainbow star (equivalently, it must be forced to some colour), see Lemma 3.5. Therefore, it will be sufficient to make sure that our partial colourings do not extend to a constant proportion of the boosters in $\mathcal {B}_Z$ .

We say that $A \in \mathcal {H}$ is an interacting edge between two sets if A is contained in their union and intersects both of them. Note that a proper partial colouring $\psi _0$ cannot be extended to B if and only if, for every proper colouring $\varphi $ of B, some edge is monochromatic under $\psi _0 \cup \varphi $ . This monochromatic edge cannot be fully contained in either Z or B, because both $\psi _0$ and $\varphi $ are proper, and is therefore an interacting edge.

Define the interface of Z with B to be the setFootnote ⁴

We will say that a partial colouring $\psi _0$ of Z and a colouring $\varphi $ of B are consistent if no edge that is interacting between Z and B is monochromatic under $\psi _0 \cup \varphi $ . In this language, a partial colouring $\psi _0$ of Z can be extended to B precisely when $\psi _0$ is consistent with some proper colouring $\varphi $ of B. Note that the interface $I(B, Z)$ contains all the information about $\psi _0$ that is needed to determine whether this is the case.

We say that is a restricted colouring of Z if, for every $z \in Z$ , there is some colour for which $(z, i) \in C$ . Recall that, in this context, we identify colourings $\psi $ of Z with the restricted colouring $\{(z,\psi (z)) \colon z \in Z \}$ . Given a restricted colouring C, we define its determined colouring to be the partial colouring $\psi _C$ , where $\psi _C(z) = i$ if every proper colouring $\psi \subseteq C$ has $\psi (z) = i$ (equivalently, if is the unique colour such that $(z,i) \in C$ ).

As we wrote in the outline, we will first define a hypergraph $\mathcal {T}$ on the vertex set in such a way that proper colourings of Z will be independent in $\mathcal {T}$ . We will then let the family $\Psi $ of partial colourings be the determined colourings of a family of (restricted colourings that are) containers for the independent sets of $\mathcal {T}$ . The edges of $\mathcal {T}$ will correspond to colourings of interfaces $I(B,Z)$ , with $B \in \mathcal {B}_Z$ , that are consistent with some proper colouring of B. In particular, we need to exercise control over the interfaces that Z has with the boosters in order to invoke the container lemma in an efficient way.

The rest of the section is organised as follows. We first prove that Z is disjoint from almost all boosters in $\mathcal {B}_Z$ and that the interface between Z and these boosters is bounded in size and well-behaved. Next, we will formally define the hypergraph $\mathcal {T}$ whose containers provide the family of partial colourings satisfying the first two properties. Finally, we will bound the degree sequence of $\mathcal {T}$ in order show that the family of containers, and therefore partial colourings, is small.

5.3 Active boosters and interactions

Let $\mathcal {B}_Z'$ be the hypergraph obtained from $\mathcal {B}_Z$ by retaining only those $B \in \mathcal {B}_Z$ that satisfy all of the following:

1. (B1) The sets B and Z are disjoint.

2. (B2) Every nonempty set in $I(B,Z)$ has $s-1$ elements.

3. (B3) The sets in $I(B,Z)$ are pairwise disjoint.Footnote ⁵

4. (B4) The family $I(B,Z)$ contains at most L sets.

The following two lemmas show that $\mathcal {B}_Z \setminus \mathcal {B}_Z'$ is small for a typical $Z \sim V_p$ .

Finally, let

$$\begin{align*}\mathcal{Z}' = \big\{Z \in \mathcal{Z} : e(\mathcal{B}_Z') \geqslant (\varepsilon/2) \cdot e(\mathcal{B})\big\}. \end{align*}$$

By Lemmas 5.1 and 5.2 and since $\lambda = \varepsilon ^2/8$ , we have, for $Z \sim V_p$ ,

$$\begin{align*}\begin{aligned} {\mathbb{P}}(Z \in \mathcal{Z}') & \geqslant {\mathbb{P}}(Z \in \mathcal{Z}) - {\mathbb{P}}\big(e(\mathcal{B}_Z \setminus \mathcal{B}_Z') \geqslant (\varepsilon/2) \cdot e(\mathcal{B})\big) \\ & \geqslant \varepsilon - \frac{\mathbb{E}\big[e(\mathcal{B}_Z \setminus \mathcal{B}_Z')\big]}{(\varepsilon/2) \cdot e(\mathcal{B})} \geqslant \varepsilon - 4\lambda/\varepsilon \geqslant \varepsilon/2. \end{aligned} \end{align*}$$

Proof of Lemma 5.1.

Suppose that $Z \sim V_p$ and let $X_1$ , $X_2$ , and $X_3$ denote the numbers of sets B that fail conditions (B1), (B2), and (B3), respectively. We have

$$\begin{align*}\mathbb{E}[X_1] = \sum_{B \in \mathcal{B}} {\mathbb{P}}(B \cap Z \neq \emptyset) \leqslant \sum_{B \in \mathcal{B}} \mathbb{E}[|B \cap Z|] = \sum_{v \in V} p \deg_{\mathcal{B}}(v) \leqslant v(\mathcal{H}) \cdot p\Delta_1(\mathcal{B}) \ll e(\mathcal{B}), \end{align*}$$

since $p \ll 1$ and $\Delta _1(\mathcal {B}) = O\big (e(\mathcal {B})/v(\mathcal {H})\big )$ . Since $X_2$ is at most the number of pairs $(A,B) \in \mathcal {H} \times \mathcal {B}$ such that $A \setminus B \subseteq Z$ and $1 \leqslant |A \setminus B| \leqslant s-2$ , we have

$$\begin{align*}\mathbb{E}[X_2] \leqslant \sum_{B \in \mathcal{B}} \sum_{t=1}^{s-2} \binom{|B|}{s-t} \cdot \Delta_{s-t}(\mathcal{H}) \cdot p^t \leqslant e(\mathcal{B}) \cdot \sum_{r=2}^{s-1} 2^K \cdot \Delta_r(\mathcal{H}) \cdot p_{\mathcal{H}}^{s-r}, \end{align*}$$

where we used the fact that every edge of $\mathcal {B}$ contains at most K vertices. Since $\mathcal {H}$ is nonclustered, we have, for every $r \in \{2, \dotsc , s-1\}$ ,

$$\begin{align*}\Delta_r(\mathcal{H}) \cdot p_{\mathcal{H}}^{s-r} \ll p_{\mathcal{H}}^{s-1} \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})} = 1, \end{align*}$$

implying that $\mathbb {E}[X_2] \ll e(\mathcal {B})$ . Finally, we can bound $X_3 - X_2$ by the number of triples $(A,A',B) \in \mathcal {H}^2 \times \mathcal {B}$ such that $A \cup A' \subseteq B \cup Z$ , $1 \leqslant |(A \cap A') \setminus B| \leqslant s-2$ , and $|A \cap B| = |A' \cap B| = 1$ . We therefore get the following upper bound by counting the options for picking B, then A, then the size of the intersection $(A \cap A') \setminus B$ , and finally $A'$ :

$$\begin{align*}\mathbb{E}[X_3 - X_2] \leqslant \sum_{B \in \mathcal{B}} |B| \cdot \Delta_1(\mathcal{H}) \cdot \sum_{r=1}^{s-2} \binom{s-1}{r} \cdot |B| \cdot \Delta_{r+1}(\mathcal{H}) \cdot p_{\mathcal{H}}^{2s-2-r}. \end{align*}$$

Using our assumptions that every edge of $\mathcal {B}$ contains at most K vertices, $\Delta _1(\mathcal {H}) \leqslant O\big ( e(\mathcal {H}) / v(\mathcal {H}) \big )$ , and $\Delta _{r+1}(\mathcal {H}) \ll p_{\mathcal {H}}^r \cdot e(\mathcal {H}) / v(\mathcal {H})$ for all $r \in \{1, \dotsc , s-2\}$ , we have

$$\begin{align*}\mathbb{E}[X_3 - X_2] \ll e(\mathcal{B}) \cdot p_{\mathcal{H}}^{2s-2} \cdot \left(\frac{e(\mathcal{H})}{v(\mathcal{H})}\right)^2 = e(\mathcal{B}). \end{align*}$$

The proof of the lemma is now complete.

Proof of Lemma 5.2.

For $B \in \mathcal {B}$ and $Z \subseteq V$ , let $i(B,Z)$ be the largest integer $\ell $ such that there are $A_1, \dotsc , A_\ell \in \mathcal {H}$ satisfying:

1. (i) $A_1 \cup \dotsb \cup A_\ell \subseteq B \cup Z$ ,

2. (ii) $|A_i \cap B| = 1$ for every ,

3. (iii) $A_1 \setminus B, \dotsc , A_\ell \setminus B$ are pairwise disjoint.

Observe that, if B satisfies (B2) and (B3), then $|I(B,Z)| = i(B,Z)$ . In particular, the assertion of the lemma will follow if we show that ${\mathbb {P}}\big (i(B,Z)> L\big ) \leqslant \lambda $ .

To this end, for a positive integer $\ell $ , let $\mathcal {A}_\ell $ denote the collection of all sequences $A_1, \dotsc , A_\ell \in \mathcal {H}$ that satisfy (ii) and (iii) above and note that

$$\begin{align*}|\mathcal{A}_\ell| \leqslant \big(|B| \cdot \Delta_1(\mathcal{H})\big)^\ell \leqslant \big(K \cdot \Delta_1(\mathcal{H})\big)^\ell. \end{align*}$$

Since, for every $(A_1, \dotsc , A_\ell ) \in \mathcal {A}_\ell $ , the set $(A_1 \cup \dotsb \cup A_\ell ) \setminus B$ has precisely $(s-1)\ell $ elements, we have

$$\begin{align*}{\mathbb{P}}\big(A_1 \cup \dotsb \cup A_\ell \subseteq B \cup Z\big) = p^{(s-1)\ell}. \end{align*}$$

Let $X_\ell $ be the number of sequences in $\mathcal {A}_\ell $ that satisfy (i). Since each such sequence has distinct coordinates and (i)–(iii) are invariant under any permutation of the sequence $(A_1, \dotsc , A_\ell )$ , we may conclude that

$$\begin{align*}{\mathbb{P}}\big(i(B, V_p) \geqslant \ell\big) \leqslant \frac{\mathbb{E}[X_\ell]}{\ell!} = \frac{|\mathcal{A}_\ell| \cdot p^{(s-1)\ell}}{\ell!} \leqslant \frac{\big(K \cdot \Delta_1(\mathcal{H}) \cdot p^{s-1}\big)^\ell}{\ell!}. \end{align*}$$

The assertion of the lemma follows because our assumptions imply that

$$\begin{align*}K \cdot \Delta_1(\mathcal{H}) \cdot p^{s-1} \leqslant K \cdot K_{\mathcal{H}}^s \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})} \cdot p_{\mathcal{H}}^{s-1} = K \cdot K_{\mathcal{H}}^s, \end{align*}$$

see (5).

5.4 The hypergraph of transversals

We will now formally define the hypergraph $\mathcal {T}$ . Using the terminology of Section 5.2, our goal is the following: For every restricted colouring C, if its determined colouring $\psi _C$ is consistent with some proper colouring of a booster $B \in \mathcal {B}^{\prime }_Z$ , then this fact will be evidenced by an edge of $\mathcal {T}$ inside C.

Let us discuss what that should mean. Suppose that $\varphi $ is a proper colouring of a booster $B \in \mathcal {B}^{\prime }_Z$ . We will show that, for every restricted colouring C, its determined colouring $\psi _C$ is consistent with $\varphi $ if and only if there is a colouring that is consistent with $\varphi $ and further satisfies $\psi _{C,\varphi } \subseteq C$ . Since $\bigcup I(B,Z)$ contains at most $(s-1) \cdot L$ vertices, see (B2), these colourings $\psi _{C,\varphi }$ can serve as the kind of witnesses we are looking for. As a result, one could actually stop here, and take the edges of the hypergraph to be all such sets $\big \{(z, \psi _{C,\varphi }(z)) \colon z \in \bigcup I(B,Z)\big \}$ . However, we will carry on and find witnesses that are minimal/irredundant.

To this end, suppose that an edge A interacting between B and Z is not monochromatic. Obviously, this means that two of its elements are coloured differently. However, we will use the fact that A intersects B in exactly one vertex (see (B2)) to conclude that either:

1. (1) $\psi $ coloured two of the vertices of $A \setminus B$ in different colours, or

2. (2) $\psi $ coloured some vertex of $A \setminus B$ in a colour different from $\varphi (A \cap B)$ .

One might be tempted to ignore the first case, which seems superfluous. Indeed, even if two vertices are coloured in different colours, one of these colours must be different from the one in $\varphi (A \cap B)$ . However, for a given $T \in I(B,Z)$ , there can be many edges $A \in \mathcal {H}$ with $A \setminus B = T$ and picking a colour that is different from the one in $\varphi (A \cap B)$ for every such A might not always be possible.

We will define the edges of $\mathcal {T}$ using transversals of $I(B,Z)$ . Given a proper colouring $\varphi $ of B, we will pick one of two obstructions for every $T \in I(B,Z)$ : either two coloured vertices $(z,c)$ , $(z',c')$ , where $z,z' \in T$ and $c \neq c'$ , or just one coloured vertex $(z,c)$ , where $z \in T$ and c different from the colour in $\varphi (A \cap B)$ for every $A \in \mathcal {H}$ such that $A \setminus B = T$ .

We now define the hypergraph $\mathcal {T}$ formally. For every $(s-1)$ -element set $T \subseteq Z$ , let

and note that $I(B,Z)$ comprises precisely those sets $T \subseteq Z \setminus B$ for which $|N_{\mathcal {H}}(T;B)| \geqslant 1$ . In particular, given $\psi $ and $\varphi $ , there is an element $T \in I(B,Z)$ that $\psi $ colours monochromatically from the colours in $\varphi \big (N_{\mathcal {H}}(T;B)\big )$ .

Definition 5.3. We let $\mathcal {T}$ be the (multi)hypergraph with vertex set whose (multi)set of edges is defined as follows. For every:

• • active booster $B \in \mathcal {B}_Z'$ ,

• • proper colouring of $\mathcal {H}[B]$ ,

• • set $I' \subseteq I(B,Z)$ satisfying , and

• • disjoint transversalsFootnote ⁶ $\{v_T : T \in I(B,Z)\}$ of $I(B,Z)$ and $\{v_T' : T \in I'\}$ of $I'$ , where $v_T \in T$ for every $T \in I(B,T)$ and $v_T' \in T$ for every $T \in I'$ ,

we add to $\mathcal {T}$ all edges of the form

$$\begin{align*}\big\{(v_T, c_T) : T \in I(B,Z)\big\} \cup \big\{(v_T',c_T') : T \in I'\big\}, \end{align*}$$

where:

• • for each $T \in I(B,Z) \setminus I'$ , the colour $c_T$ is an arbitrary element of ;Footnote ⁷

• • for each $T \in I'$ , the colours $c_T$ and $c_T'$ are two arbitrary, distinct elements of .

An immediate consequence of the definition should be that every proper colouring $\psi $ of Z, when viewed as a restricted colouring, is an independent set of $\mathcal {T}$ . This is because the determined colouring of $\psi $ , which is just $\psi $ itself, cannot be extended to any $B \in \mathcal {B}_Z'$ .

Proof. Suppose that some proper colouring $\psi $ of $\mathcal {H}[Z]$ contains an edge of $\mathcal {T}$ . This means that there are an active booster $B \in \mathcal {B}_Z'$ , a proper colouring $\varphi $ of $\mathcal {H}[B]$ , a set $I' \subseteq I(B,Z)$ , and disjoint transversals $\{v_T : T \in I(B,Z)\}$ and $\{v_T' : T \in I'\}$ such that, for every $T \in I(B,Z)$ , either $\psi (v_T) \notin \varphi \big (N_{\mathcal {H}}(T;B)\big )$ or $\psi (v_T) \neq \psi (v_T')$ . In particular, there is no $T \in I(B,Z)$ whose all elements receive the same colour from the set $\varphi \big (N_{\mathcal {H}}(T;B)\big )$ . On the other hand, since B is disjoint from Z, see (B1), one may naturally define the colouring . Since B is an active booster, this colouring $\psi \cup \varphi $ is not a proper colouring of $\mathcal {H}[Z \cup B]$ , which means that there is an $A \in \mathcal {H}[Z \cup B]$ that $\psi \cup \varphi $ makes monochromatic. Since $\psi $ and $\varphi $ are proper colourings of $\mathcal {H}[Z]$ and of $\mathcal {H}[B]$ , respectively, every such A must have a nonempty intersection with both Z and B. Our choice of $\mathcal {B}_Z'$ , see (B2), implies that $|A \cap B| = 1$ and, consequently, that $A \setminus B \in I(B,Z)$ . Moreover,

$$\begin{align*}(\psi\cup\varphi)(A) = \psi(A \setminus B) = \varphi(A \cap B), \end{align*}$$

which means that all elements of $A \setminus B$ receive the same colour as the unique vertex in $A \cap B$ , which belongs to $N_{\mathcal {H}}(A \setminus B;B)$ , a contradiction.

More importantly, the definition implies that, for every restricted colouring C that induces a small number of edges in $\mathcal {T}$ , its determined colouring $\psi _C$ forces a constant fraction of the vertices of $\mathcal {H}$ to some colour. Recall that we denote the set of vertices that $\psi _C$ forces to some colour by $F(\psi _C)$ .

Lemma 5.5. If is a restricted colouring, then

$$\begin{align*}e(\mathcal{T}[C]) \geqslant e(\mathcal{B}_Z') - |F(\psi_C)| \cdot \Delta_1(\mathcal{B}). \end{align*}$$

5.5 The distribution of edges of $\mathcal {T}$

The key step in the proof of Theorem 5 will be to construct a small family of containers for proper -colourings of $\mathcal {H}[Z]$ using the information provided by the hypergraph $\mathcal {T}$ . In order to do this, we will first show that, for a typical choice of $Z \sim V_p$ , we may find a subhypergraph $\mathcal {T}' \subseteq \mathcal {T}$ that contains almost as many edges as $\mathcal {T}$ and satisfies

$$\begin{align*}\Delta_1(\mathcal{T}') = O\left(\frac{e(\mathcal{T}')}{|Z|}\right) \qquad \text{and} \qquad \Delta_2(\mathcal{T}') = o\left(\frac{e(\mathcal{T}')}{|Z|}\right). \end{align*}$$

We will then split $\mathcal {T}'$ into uniform hypergraphs $\mathcal {T}_1, \dotsc , \mathcal {T}_{2L}$ , where, for each , the u-uniform hypergraph $\mathcal {T}_u$ comprises all edges of $\mathcal {T}'$ of cardinality u. Finally, we will apply the hypergraph container lemma, Theorem B, to each of the $\mathcal {T}_u$ in order to build containers for independent sets of $\mathcal {T}$ : the container for an independent set I will be the intersection of the $2L$ containers for I in the hypergraphs $\mathcal {T}_1, \dotsc , \mathcal {T}_{2L}$ . The assumptions on $\Delta _1(\mathcal {T}')$ and $\Delta _2(\mathcal {T}')$ will guarantee that there are only $\exp (o(|Z|))$ different containers. Lemma 5.5 provides a useful, structural description of all such containers.

In order to guarantee the existence of a $\mathcal {T}' \subseteq \mathcal {T}$ with the required properties, we shall estimate the $\ell ^2$ -norm of the sequences of vertex degrees and vertex-pair degrees of $\mathcal {T}$ and invoke the following simple proposition.

Proposition 5.6. Suppose that $\mathcal {G}$ is a (multi)hypergraph with vertex set V. For all positive integers t and m, there is a subhypergraph $\mathcal {G}' \subseteq \mathcal {G}$ with $e(\mathcal {G}') \geqslant e(\mathcal {G}) - m$ and

$$\begin{align*}\Delta_t(\mathcal{G}') \leqslant \frac{1}{m} \cdot \sum_{T \in \binom{V}{t}} \deg_{\mathcal{G}}(T)^2. \end{align*}$$

Proof. We may assume that $m \leqslant e(\mathcal {G})$ ; indeed, if $m> e(\mathcal {G})$ , we may simply take $\mathcal {G}'$ to be the empty hypergraph. We obtain $\mathcal {G}'$ from $\mathcal {G}$ by iteratively removing m edges that contain some set $T \in \binom {V}{t}$ with largest degree. Since each of the m removed edges contained a t-element subset of degree at least $\Delta _t(\mathcal {G}')$ , we have

$$\begin{align*}\sum_{T \in \binom{V}{t}} \deg_{\mathcal{G}}(T)^2 \geqslant m \cdot \Delta_t(\mathcal{G}'), \end{align*}$$

as claimed.

The following two crucial lemmas give upper bounds on the expectations of the $\ell ^2$ -norms of the sequences of vertex degrees and vertex-pair degrees of $\mathcal {T}$ .

Lemma 5.7. If $Z \sim V_p$ , then

$$\begin{align*}\mathbb{E}\left[\sum_{v \in V(\mathcal{T})} \deg_{\mathcal{T}}(v)^2\right] \leqslant \frac{\Gamma \cdot e(\mathcal{B})^2}{p_{\mathcal{H}} \cdot v(\mathcal{H})}, \end{align*}$$

where $\Gamma $ is a constant that depends only on K, $K_{\mathcal {H}}$ , L, r, and s.

Lemma 5.8. If $Z \sim V_p$ , then

$$\begin{align*}\mathbb{E}\left[\sum_{T \in \binom{V(\mathcal{T})}{2}} \deg_{\mathcal{T}}(T)^2\right] \leqslant \frac{\sigma \cdot e(\mathcal{B})^2}{p_{\mathcal{H}} \cdot v(\mathcal{H})} \end{align*}$$

for some $\sigma = o(1)$ .

Proof of Lemma 5.7.

For every $v \in V$ , define

We claim that, for all $v \in Z$ and ,

(6) $$ \begin{align} \deg_{\mathcal{T}}(v,i) \leqslant \gamma(v) \cdot \Delta_1(\mathcal{B}) \cdot (rs)^{2L+K}, \end{align} $$

where $\deg _{\mathcal {T}}$ counts edges with multiplicities. Indeed, if $v \in Z$ , then $\gamma (v) \cdot \Delta _1(\mathcal {B})$ is an upper bound on the number of $B \in \mathcal {B}_Z'$ such that $v \in T$ for some $T \in I(B,Z)$ . Every such B gives rise to at most

$$\begin{align*}r^K \cdot \big((s-1)r\big)^L \cdot \sum_{i=0}^L \big((s-2)r\big)^{i} \leqslant (rs)^{2L+K} \end{align*}$$

edges of $\mathcal {T}$ , counting multiplicities; indeed, there are at most $r^K$ proper -colourings of B, at most $(s-1)r$ choices for the pair $(v_T,c_T)$ for every $T \in I(B,Z)$ , and at most $(s-2)r$ further choices for the pair $(v_T', c_T')$ for every $T \in I' \subseteq I(B,Z)$ , see Definition 5.3. Moreover, all edges of $\mathcal {T}$ that contain $(v,i)$ must come from one such set B.

Since , inequality (6) implies that

(7) $$ \begin{align} \begin{aligned} \sum_{v \in V(\mathcal{T})} \deg_{\mathcal{T}}(v)^2 & \leqslant r \cdot \sum_{v \in Z} \gamma(v)^2 \cdot \Delta_1(\mathcal{B})^2 \cdot (rs)^{4L+2K} \\ & \leqslant \frac{K \cdot e(\mathcal{B})^2}{v(\mathcal{H})^2} \cdot (rs)^{4L+2K+1} \cdot \sum_{v \in Z} \gamma(v)^2. \end{aligned} \end{align} $$

In the remainder of the proof, we will bound the expected value of the sum in the right-hand side of (7).

Fix an arbitrary $v \in V$ and observe that

$$\begin{align*}\mathbb{E}[\gamma(v)^2] = \sum_{\substack{A, A' \in \mathcal{H} \\ v \in A \cap A'}} \underbrace{{\mathbb{P}}\big(|A \cap (Z \setminus \{v\})| = s-2 \text{ and } |A' \cap (Z \setminus \{v\})| = s-2\big)}_{P(A,A')}; \end{align*}$$

note that $P(A,A) \leqslant (s-1) \cdot p^{s-2}$ and that, if $A \neq A'$ , then

$$\begin{align*}P(A,A') \leqslant (s-1)^2 \cdot p^{|A \cup A'| - 3} = (s-1)^2 \cdot p^{2s-3-|A \cap A'|}. \end{align*}$$

Consequently,

$$\begin{align*}\mathbb{E}[\gamma(v)^2] = \deg_{\mathcal{H}}(v) \cdot (s-1) \cdot p^{s-2} \cdot \left(1 + \sum_{t=1}^{s-1} \binom{s-2}{t-1} \cdot (s-1) \cdot \Delta_t(\mathcal{H})\cdot p^{s-1-t} \right). \end{align*}$$

Our assumption that $\mathcal {H}$ is nonclustered implies that, for every ,

$$\begin{align*}\Delta_t(\mathcal{H}) \cdot p^{s-1-t} \leqslant K_{\mathcal{H}}^{s-1} \cdot \Delta_t(\mathcal{H}) \cdot p_{\mathcal{H}}^{s-1-t} \leqslant K_{\mathcal{H}}^s \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})} \cdot p_{\mathcal{H}}^{s-2} = \frac{K_{\mathcal{H}}^s}{p_{\mathcal{H}}}. \end{align*}$$

and thus

$$\begin{align*}\mathbb{E}[\gamma(v)^2] \leqslant \Gamma \cdot \deg_{\mathcal{H}}(v) \cdot p_{\mathcal{H}}^{s-3} \end{align*}$$

for some constant $\Gamma $ that depends only on $K_{\mathcal {H}}$ and s. Since the variable $\gamma (v)$ is independent of the event $v \in Z$ , we have

$$\begin{align*}\begin{aligned} \mathbb{E}\left[\sum_{v \in Z} \gamma(v)^2\right] & = \sum_{v \in V} p \cdot \mathbb{E}[\gamma(v)^2] \leqslant \Gamma \cdot K_{\mathcal{H}} \cdot p_{\mathcal{H}}^{s-2} \cdot \sum_{v \in V} \deg_{\mathcal{H}}(v) \\ & = \Gamma \cdot K_{\mathcal{H}} \cdot p_{\mathcal{H}}^{s-2} \cdot s \cdot e(\mathcal{H}) = \frac{\Gamma \cdot K_{\mathcal{H}} \cdot s \cdot v(\mathcal{H})}{p_{\mathcal{H}}}. \end{aligned} \end{align*}$$

Taking expectations of both sides of (7) and substituting the above inequality yields the assertion of the lemma.

Proof of Lemma 5.8.

For every pair of distinct $v, w \in V$ , let $\gamma _2(v, w)$ be the number of triples $(A_v, A_w, B) \in \mathcal {H}^2 \times \mathcal {B}$ satisfying the following:

1. (i) $v \in A_v \setminus B$ and $w \in A_w \setminus B$ ,

2. (ii) $|A_v \cap B| = |A_w \cap B| = 1$ ,

3. (iii) $A_v \cap A_w \subseteq B$ or $A_v \setminus B = A_w \setminus B$ ,

4. (iv) $(A_v \cup A_w) \setminus (B \cup \{v, w\}) \subseteq Z$ .

We claim that, for every pair of distinct $v, w \in Z$ and all ,

(8) $$ \begin{align} \deg_{\mathcal{T}}\{(v,i), (w,j)\} \leqslant \gamma_2(v,w) \cdot (rs)^{2L+K}. \end{align} $$

Indeed, if $v,w \in Z$ , then $\gamma _2(v,w)$ is an upper bound on the number of $B \in \mathcal {B}_Z'$ such that $v \in T_v$ and $w \in T_w$ for some $T_v, T_w \in I(B,Z)$ . Every such B gives rise to at most $(rs)^{2L+K}$ edges of $\mathcal {T}$ , counting multiplicities, and all edges of $\mathcal {T}$ that contain both $(v,i)$ and $(w,j)$ must come from one such set B.

Since and no edge of $\mathcal {T}$ contains a pair of vertices $\{(v,i), (v,j)\}$ with $i \neq j$ , inequality (8) implies that

(9) $$ \begin{align} \sum_{T \in \binom{V(\mathcal{T})}{2}} \deg_{\mathcal{T}}(T)^2 \leqslant r^2 \cdot \sum_{v,w \in Z} \gamma_2(v,w)^2 \cdot (rs)^{4L+2K}. \end{align} $$

In the remainder of the proof, we will bound the expected value of the sum in the right-hand side of (9).

Claim 5.9. For every pair of distinct vertices $v, w \in V$ ,

$$\begin{align*}\mathbb{E}[\gamma_2(v, w)^2] \ll \frac{e(\mathcal{B})}{p_{\mathcal{H}} \cdot v(\mathcal{H})} \cdot \mathbb{E}[\gamma_2(v,w)]. \end{align*}$$

Proof. Let $\mathcal {X}$ denote the family of all triples $(A_v, A_w, B)$ that satisfy i–iii above and observe that

see (iv). Therefore, it suffices to show that, for each $(A_v, A_w, B) \in \mathcal {X}$ ,

(10) $$ \begin{align} \mathbb{E}\big[\gamma_2(v,w) \mid (A_v \cup A_w) \setminus (B \cup \{v,w\}) \subseteq Z\big] \ll \frac{\Delta_1(\mathcal{B})}{p_{\mathcal{H}}}. \end{align} $$

We partition the family $\mathcal {X}$ according to the intersection pattern with our chosen triple $(A_v, A_w, B)$ . First, for every $t \in \{0, \dotsc , s-3\}$ , denote by $\mathcal {X}_t$ the family of all triples $(A_v', A_w', B')$ such that $A_v' \setminus B = A_w' \setminus B$ and $A_v'$ intersects $(A_v \cup A_w) \setminus (B \cup \{v,w\})$ in exactly t elements. Second, for every $(t_v, t_w) \in \{0, \dotsc , s-2\}^2$ , denote by $\mathcal {X}_{t_v, t_w}$ the family of all triples $(A_v', A_w', B')$ such that $A_v' \cap A_w' \subseteq B$ and, for $u \in \{v,w\}$ , the set $A_u'$ intersects $(A_v \cup A_w) \setminus (B \cup \{v,w\})$ in exactly $t_u$ elements. Observe that

$$\begin{align*}\mathcal{X} = \bigcup_{t=0}^{s-3} \mathcal{X}_t \cup\bigcup_{t_v, t_w = 0}^{s-2} \mathcal{X}_{t_v,t_w}. \end{align*}$$

Further, note that, for every $t \in \{0, \dotsc , s-3\}$ ,

$$\begin{align*}|\mathcal{X}_t| \leqslant \Delta_{t+2}(\mathcal{H}) \cdot \Delta_1(\mathcal{B}) \cdot \max_{B \in \mathcal{B}} |B| \ll p_{\mathcal{H}}^{t+1} \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})} \cdot \Delta_1(\mathcal{B}) = p_{\mathcal{H}}^{t+2-s} \cdot \Delta_1(\mathcal{B}). \end{align*}$$

Crucially, we claim that, for every $t_v, t_w \in \{0, \dotsc , s-2\}$ ,

$$\begin{align*}\begin{aligned} |\mathcal{X}_{t_v, t_w}| & \leqslant \min\big\{ \Delta_{t_v+1}(\mathcal{H}) \cdot \Delta_{t_w+2}(\mathcal{H}), \Delta_{t_w+1}(\mathcal{H}) \cdot \Delta_{t_v+2}(\mathcal{H}) \big\} \cdot \Delta_1(\mathcal{B}) \\ & \ll p_{\mathcal{H}}^{t_v+t_w+1} \cdot \left(\frac{e(\mathcal{H})}{v(\mathcal{H})}\right)^2 \cdot \Delta_1(\mathcal{B}) = p_{\mathcal{H}}^{t_v+t_w-2s+3} \cdot \Delta_1(\mathcal{B}). \end{aligned} \end{align*}$$

Indeed, $\Delta _t(\mathcal {H}) \leqslant K p_{\mathcal {H}}^{t-1} \cdot e(\mathcal {H}) / v(\mathcal {H})$ for all and $\Delta _t(\mathcal {H}) \ll p_{\mathcal {H}}^{t-1} \cdot e(\mathcal {H}) / v(\mathcal {H})$ when $2 \leqslant t \leqslant s-1$ and one of $\max \{t_v, t_w\} + 1$ or $\min \{t_v, t_2\} + 2$ must belong to $\{2, \dotsc , s-1\}$ . Since $p = O(p_{\mathcal {H}})$ , estimate (10) follows after noting that, first, for every $t \in \{0, \dotsc , s-3\}$ and every $(A_v', A_w', B') \in \mathcal {X}_t$ ,

$$\begin{align*}{\mathbb{P}}\big((A_v' \cup A_w') \setminus (B' \cup \{v,w\}) \subseteq Z \mid (A_v \cup A_w) \setminus (B \cup \{v,w\}) \subseteq Z\big) = p^{s-3-t}, \end{align*}$$

and, second, for all $(t_v, t_w) \in \{0, \dotsc , s-2\}^2$ and every $(A_v', A_w', B') \in \mathcal {X}_{t_v, t_w}$ ,

$$\begin{align*}{\mathbb{P}}\big((A_v' \cup A_w') \setminus (B' \cup \{v,w\}) \subseteq Z \mid (A_v \cup A_w) \setminus (B \cup \{v,w\}) \subseteq Z\big) = p^{2s-4-t_v-t_w}. \end{align*}$$

Since $p \leqslant K_{\mathcal {H}} \cdot p_{\mathcal {H}}$ and $\Delta _1(\mathcal {B}) \leqslant K \cdot e(\mathcal {B}) / v(\mathcal {H})$ , this concludes the proof of the claim.

Claim 5.10. We have

$$\begin{align*}\mathbb{E}\left[\sum_{v,w \in Z} \gamma_2(v,w) \right] = O\big(e(\mathcal{B})\big). \end{align*}$$

Proof. Let $\mathcal {X}_1$ denote the family of all pairs $(A,B) \in \mathcal {H} \times \mathcal {B}$ such that $|A \cap B| = 1$ and let $\mathcal {X}_2$ denote the family of all triples $(A', A", B') \in \mathcal {H}^2 \times \mathcal {B}$ such that $|A' \cap B'| = |A" \cap B'| = 1$ and $A' \cap A" \subseteq B'$ . Note that

(11) $$ \begin{align} \sum_{v,w \in Z} \gamma_2(v,w) = \binom{s-1}{2} & \cdot \left|\big\{(A, B) \in \mathcal{X}_1 : A \setminus B \subseteq Z \big\}\right| \nonumber\\ & + (s-1)^2 \cdot \left|\big\{(A', A", B') \in \mathcal{X} : (A' \cup A") \setminus B' \subseteq Z \big\}\right|. \end{align} $$

The claim now follows as

$$ \begin{align*} |\mathcal{X}_1| & \leqslant \sum_{B \in \mathcal{B}} |B| \cdot \Delta_1(\mathcal{H}) = O\left(e(\mathcal{B}) \cdot \frac{e(\mathcal{H})}{v(\mathcal{H})}\right), \\ |\mathcal{X}_2| & \leqslant \sum_{B \in \mathcal{B}} |B|^2 \cdot \Delta_1(\mathcal{H})^2 = O\left( e(\mathcal{B}) \cdot \frac{e(\mathcal{H})^2}{v(\mathcal{H})^2}\right), \end{align*} $$

and, for all $(A,B) \in \mathcal {X}_1$ and $(A', A", B') \in \mathcal {X}_2$ ,

$$ \begin{align*} {\mathbb{P}}\big(A \setminus B \subseteq Z\big) & = p^{s-1} = O\big(p_{\mathcal{H}}^{s-1}\big), \\ {\mathbb{P}}\big((A' \cup A") \setminus B' \subseteq Z\big) & = p^{2s-2} = O\big(p_{\mathcal{H}}^{2s-2}\big). \end{align*} $$

Indeed, since $p_{\mathcal {H}}^{s-1} \cdot e(\mathcal {H}) = v(\mathcal {H})$ , taking expectations of both sides of (11) and substituting the above estimates yields the claimed estimate.

Since the variable $\gamma _2(v,w)$ is independent of the event $\{v,w\} \subseteq Z$ , we have

$$\begin{align*}\begin{aligned} \mathbb{E}\left[\sum_{v,w \in Z} \gamma_2(v,w)^2\right] & = p^2 \cdot \sum_{v,w \in V} \mathbb{E}[\gamma_2(v,w)^2] \ll p^2 \cdot \frac{e(\mathcal{B})}{p_{\mathcal{H}} \cdot v(\mathcal{H})}\cdot \sum_{v,w \in V} \mathbb{E}[\gamma_2(v,w)] \\ & = \frac{e(\mathcal{B})}{p_{\mathcal{H}} \cdot v(\mathcal{H})} \cdot \mathbb{E}\left[\sum_{v,w \in Z} \gamma_2(v,w)\right] = O\left(\frac{e(\mathcal{B})^2}{p_{\mathcal{H}} \cdot v(\mathcal{H})}\right), \end{aligned} \end{align*}$$

where the first inequality follows from Claim 5.9 and the second inequality follows from Claim 5.10. The assertion of the lemma follows by taking expectations of both sides of (9) and substituting the above inequality.

5.6 Containers for colourings

Let $\sigma $ be the sequence from the statement of Lemma 5.8 and let $\tau = \sigma ^{1/(2L+1)} = o(1)$ . Let $\mathcal {Z}"$ be the collection of all $Z \in \mathcal {Z}'$ such that

$$\begin{align*}\sum_{v \in V(\mathcal{T})} \deg_{\mathcal{T}}(v)^2 \leqslant \Gamma_{\mathcal{H}} \cdot \frac{e(\mathcal{B})^2}{|Z|} \qquad \text{and} \qquad \sum_{T \in \binom{V(\mathcal{T})}{2}} \deg_{\mathcal{T}}(T)^2 \leqslant \tau^{2L} \cdot \frac{e(\mathcal{B})^2}{|Z|}. \end{align*}$$

Since $\mathcal {Z}" \subseteq \mathcal {Z}$ , the set Z satisfies (i) in the statement of Theorem 5.

Before we prove the lemma, let us point out that it implies the assertion of the theorem. Indeed, it follows from Lemmas 5.7 and 5.8, Markov’s inequality, and standard estimates for the tails of the binomial distribution that, when $Z \sim V_p$ ,

$$\begin{align*}\begin{aligned} {\mathbb{P}}(Z \in \mathcal{Z}") & \geqslant {\mathbb{P}}(Z \in \mathcal{Z}') - {\mathbb{P}}\big(|Z| \geqslant 2p_{\mathcal{H}} v(\mathcal{H})\big)-2\Gamma/\Gamma_{\mathcal{H}} - 2\sigma \cdot \tau^{-2L} \\ & \geqslant \varepsilon/2 - o(1) - 2\Gamma/\Gamma_{\mathcal{H}} - 2\tau \geqslant \varepsilon/4. \end{aligned} \end{align*}$$

Proof of Lemma 5.11.

Suppose that $Z \in \mathcal {Z}"$ . We will construct the desired family of partial colourings of Z by applying the container lemma, Theorem B, to a collection of uniform subhypergraphs of the hypergraph $\mathcal {T}$ .

To this end, note first that Proposition 5.6, invoked twice, implies that $\mathcal {T}$ contains a subhypergraph $\mathcal {T}'$ with at least $e(\mathcal {T}) - 2 \cdot (\varepsilon /16) \cdot e(\mathcal {B})$ edges that satisfies

$$\begin{align*}\Delta_1(\mathcal{T}') \leqslant \frac{16 \Gamma_{\mathcal{H}}}{\varepsilon} \cdot \frac{e(\mathcal{B})}{|Z|} \qquad \text{and} \qquad \Delta_2(\mathcal{T}') \leqslant \frac{16 \tau^{2L}}{\varepsilon} \cdot \frac{e(\mathcal{B})}{|Z|} \leqslant \tau^{2L-1} \cdot \frac{e(\mathcal{B})}{|Z|}. \end{align*}$$

Fix one such hypergraph $\mathcal {T}'$ and note that

(12) $$ \begin{align} e(\mathcal{T}') \leqslant r \cdot |Z| \cdot \Delta_1(\mathcal{T}') \leqslant \frac{16 \Gamma_{\mathcal{H}} r}{\varepsilon} \cdot e(\mathcal{B}). \end{align} $$

Let $\mathcal {T}_1, \dotsc , \mathcal {T}_{2L}$ be the subhypergraphs of $\mathcal {T}'$ that comprise all edges of $\mathcal {T}'$ of cardinalities $1, \dotsc , 2L$ , respectively, and note that $\mathcal {T}' = \mathcal {T}_1 \cup \dotsb \cup \mathcal {T}_{2L}$ , since every edge of $\mathcal {T}$ has cardinality at most

$$\begin{align*}2 \cdot \max_{B \in \mathcal{B}_Z'} |I(B,Z)| \leqslant 2L, \end{align*}$$

see Definition 5.3. Define

and fix an arbitrary $u \in U$ . Since

$$\begin{align*}\Delta_1(\mathcal{T}_u) \leqslant \Delta_1(\mathcal{T}') \leqslant \frac{16\Gamma_{\mathcal{H}}}{\varepsilon} \cdot \frac{e(\mathcal{B})}{|Z|} \leqslant \frac{256\Gamma_{\mathcal{H}} L r}{\varepsilon^2} \cdot \frac{e(\mathcal{T}_u)}{v(\mathcal{T}_u)} \end{align*}$$

and, for every $\ell \in \{2, \dotsc , u\}$ ,

$$\begin{align*}\Delta_\ell(\mathcal{T}_u) \leqslant \Delta_2(\mathcal{T}') \leqslant \tau^{2L-1} \cdot \frac{e(\mathcal{B})}{|Z|} \leqslant \frac{16Lr}{\varepsilon} \cdot \tau^{\ell-1} \cdot \frac{e(\mathcal{T}_u)}{v(\mathcal{T}_u)}, \end{align*}$$

we may apply Theorem B, with ,

to get an integer t (that depends only on $\varepsilon $ , $\Gamma _{\mathcal {H}}$ , r, and L) and a collection $\mathcal {C}_u$ of at most

$$\begin{align*}\left(\sum_{i=0}^{\tau r |Z|} \binom{r|Z|}{i}\right)^t \leqslant \left(\frac{er|Z|}{\tau r |Z|}\right)^{t \tau r |Z|} = \exp\big(o(|Z|)\big) \end{align*}$$

subsets of with the following properties:

1. (i) Every proper colouring of $\mathcal {H}[Z]$ , viewed as a subset of , is contained in a member of $\mathcal {C}_u$ .

2. (ii) Every member of $\mathcal {C}_u$ induces fewer than $\varepsilon /(16L) \cdot e(\mathcal {B})$ edges in $\mathcal {T}_u$ .

Finally, we let $\Psi $ to be the collection of all partial colourings $\psi _C$ , where C is a set of the form

$$\begin{align*}C = \bigcap_{u \in U} C_u, \end{align*}$$

where, for each $u \in U$ , the set $C_u$ is a member of $\mathcal {C}_u$ , such that $\pi _1(C) = Z$ ; note that

$$\begin{align*}|\Psi| \leqslant \prod_{u \in U} |\mathcal{C}_u| = \exp\big(o(|Z|)\big). \end{align*}$$

It follows from (i) that, for each $u \in U$ , every proper colouring of $\mathcal {H}[Z]$ is contained in some set $C_u \in \mathcal {C}_u$ and, consequently, in some set C of the above form; in particular $\psi $ extends the partial colouring $\psi _C \in \Psi $ . We now show that, for every partial colouring $\psi _C \in \Psi $ , we have $|F(\psi _C)| = \Omega \big (v(\mathcal {H})\big )$ ; this will conclude the proof of the lemma, as every vertex in $F(\psi _C)$ is the centre of a rainbow star.

To this end, choose an arbitrary set C of the above form. Since $e(\mathcal {T}_u[C]) \leqslant e(\mathcal {T}_u[C_u]) \leqslant \varepsilon /(16L) \cdot e(\mathcal {B})$ when $u \in U$ , see (ii) above, and $e(\mathcal {T}_u[C]) \leqslant e(\mathcal {T}_u) < \varepsilon /(16L) \cdot e(\mathcal {B})$ when $u \notin U$ , we have

$$\begin{align*}e(\mathcal{T}'[C]) = \sum_{u = 1}^{2L} e(\mathcal{T}_u[C]) \leqslant 2L \cdot \varepsilon/(16L) \cdot e(\mathcal{B}) = (\varepsilon/8) \cdot e(\mathcal{B}). \end{align*}$$

Consequently,

$$\begin{align*}e(\mathcal{T}[C]) \leqslant e(\mathcal{T}'[C]) + e(\mathcal{T}) - e(\mathcal{T}') \leqslant (\varepsilon/4) \cdot e(\mathcal{B}) \end{align*}$$

and thus Lemma 5.5 implies that

$$\begin{align*}|F(\psi_C)| \geqslant \frac{e(\mathcal{B}_Z') - e(\mathcal{T}[C])}{\Delta_1(\mathcal{B})} \geqslant \frac{\varepsilon \cdot e(\mathcal{B})}{4 \cdot \Delta_1(\mathcal{B})} \geqslant \frac{\varepsilon}{4K} \cdot v(\mathcal{H}), \end{align*}$$

since $Z \in \mathcal {Z}'$ implies that $e(\mathcal {B}_Z') \geqslant (\varepsilon /2) \cdot e(\mathcal {B})$ .