1.1 A Dirac-type theorem for perfect H-tilings

In this subsection, we state Theorem 1.8, which asymptotically resolves Problem 1.1 when $\chi _<(H)\geq 3$ . This result will depend on several definitions and parameters that we now introduce.

One can think of the interval chromatic number as the natural ordered analogue of the chromatic number of a graph. Moreover, whilst the Erdős–Stone–Simonovits theorem [Reference Erdős and Simonovits8, Reference Erdős and Stone9] asserts that $1-1/(\chi (H)-1)+o(1)$ is the edge density threshold for ensuring a copy of a graph H in G, the Erdős–Stone–Simonovits theorem for ordered graphs due to Pach and Tardos [Reference Pach and Tardos24] asserts the corresponding threshold in the ordered graph setting is $1-1/(\chi _< (H)-1)+o(1)$ .

The next definition is the relevant parameter for studying almost perfect H-tilings in graphs.

Definition 1.3 (Critical chromatic number)

The critical chromatic number $\chi _{cr}(F)$ of an unordered graph F is defined as

$$\begin{align*}\chi_{cr}(F):=(\chi(F) - 1)\frac{|F|}{|F| - \sigma(F)}, \end{align*}$$

where $\sigma (F)$ denotes the size of the smallest possible colour class in any $\chi (F)$ -colouring of F.

Note that $\chi (H)-1<\chi _{cr}(H)\leq \chi (H)$ for all graphs H, and $\chi _{cr}(H)=\chi (H)$ precisely when every $\chi (H)$ -colouring of H has colour classes of equal size.

We informally refer to an H-tiling in an n-vertex (ordered) graph G as an almost perfect H-tiling if it covers all but at most $o(n)$ vertices of G. Komlós [Reference Komlós16] proved that $(1-1/\chi _{cr}(H))n$ is the minimum degree threshold for forcing an almost perfect H-tiling in an n-vertex graph G. In fact, it was later shown [Reference Shokoufandeh and Zhao26] that such graphs G contain H-tilings covering all but a constant number of the vertices in G. In the setting of ordered graphs, a related parameter $\chi ^*_{cr}(H)$ turns out to be the relevant parameter for forcing an almost perfect H-tiling. To introduce this parameter, we need the following definitions.

For two subsets $X, Y$ of $[n]$ , we write $X<Y$ if $x<y$ for all $x\in X$ and $y\in Y$ . Let B be a complete k-partite unordered graph with parts $U_1,\ldots , U_k$ and $\sigma $ be a permutation of the set $[k]$ . An interval labelling of B with respect to $\sigma $ is a bijection $\phi : V(B)\rightarrow [|B|]$ such that $\phi (U_i)<\phi (U_j)$ if $\sigma (i)<\sigma (j)$ : that is,

$$ \begin{align*} \phi(U_{\sigma^{-1}(1)})<\cdots<\phi(U_{\sigma^{-1}(k)}). \end{align*} $$

For brevity, we will usually drop $\phi $ and just write $U_{\sigma ^{-1}(1)}<\cdots <U_{\sigma ^{-1}(k)}.$ Given $t \in \mathbb N$ , write $B(t)$ for the blow-up of B with vertex set $\bigcup _{x\in V(B)}V_x$ , where the $V_x$ s are sets of t independent vertices; so there are all possible edges between $V_x$ and $V_y$ in $B(t)$ if $xy \in E(B)$ . Given an interval labelling $\phi $ of B, let $(B(t), \phi )$ be the ordered graph obtained from $B(t)$ by equipping $V(B(t))$ with a vertex ordering, satisfying $V_x < V_y$ for every $x, y\in V(B)$ with $\phi (x)<\phi (y)$ . We refer to $(B(t), \phi )$ as an ordered blow-up of B.

Note that in Definition 1.4, we did not impose any restriction on the size of the parts of a bottlegraph. However, as we will see in Proposition 5.1, it suffices to consider bottlegraphs $B'$ , where all parts are of the same size except for perhaps one smaller part. More precisely, given any bottlegraph B of H, there is another bottlegraph $B'$ with this structure such that $\chi _{cr}(B')=\chi _{cr}(B)$ . This bottle-like structure is where the name bottlegraph is derived and was first used by Komlós [Reference Komlós16] in the setting of unordered graphs.

Definition 1.5 (Ordered critical chromatic number)

The ordered critical chromatic number $\chi ^*_{cr}(F)$ of an ordered graph F is defined as

$$ \begin{align*} \chi_{cr}^*(F):=\inf\{\chi_{cr}(B):B\text{ is a bottlegraph of }F\}. \end{align*} $$

We say a bottlegraph B of F is optimal if $\chi _{cr}(B)=\chi _{cr}^*(F)$ .

Notice that $\chi _<(H)-1 \leq \chi _{cr}^*(H)$ for all ordered graphs H as each bottlegraph B of H must have chromatic number at least $\chi _<(H)$ so $\chi _<(H)-1 < \chi _{cr}(B)$ . In fact, Proposition 2.6 in Section 2 yields a stronger lower bound on $\chi _{cr}^*(H)$ . On the other hand, (in contrast to $\chi _{cr}(F)$ for unordered graphs F), we will also see examples of ordered graphs where $\chi _{cr}^*(H)$ is much larger than $\chi _<(H)$ . Note though that $\chi _{cr}^*(H)\leq h$ for any ordered graph H on $[h]$ as $K_{h}$ is a bottlegraph of H. In fact, this upper bound is attained when H is such that $1$ and $2$ are adjacent or $h-1$ and h are adjacent; this is an immediate consequence of Proposition 11.1. To aid the reader’s intuition, in Section 3.3, we give examples of ordered graphs H, where we compute $\chi ^*_{cr}(H)$ . Various bounds on $\chi ^*_{cr}(H)$ are given in Section 11.

The next result, a simple corollary of [Reference Balogh, Li and Treglown3, Theorem 4.3], shows that $\chi ^*_{cr}(H)$ is a relevant parameter for forcing an almost perfect H-tiling in an ordered graph.Footnote ¹

Theorem 1.6 (Balogh, Li and Treglown [Reference Balogh, Li and Treglown3])

Let H be an ordered graph. Then for every $\eta>0$ , there exists an integer $n_0=n_0(H,\eta )$ so that every ordered graph G on $n\geq n_0$ vertices with

$$ \begin{align*} \delta(G)\geq\left(1-\frac{1}{\chi^*_{cr}(H)}\right)n \end{align*} $$

contains an H-tiling covering all but at most $\eta n$ vertices.

At first sight, it is not clear if the minimum degree threshold in Theorem 1.6 is best possible. However, Theorem 12.1 in Section 12 shows that Theorem 1.6 is best possible in the sense that one cannot replace the $1-{1}/{\chi ^*_{cr}(H)}$ term in the minimum degree condition with any other fixed constant term $a< 1-{1}/{\chi ^*_{cr}(H)}$ . Thus, Theorem 12.1 and Theorem 1.6 provide an analogue of Komlós’ theorem in the ordered setting.

Unusually, in the proof of Theorem 12.1, for most ordered graphs H, we do not simply produce an explicit extremal example. Indeed, if one has not explicitly computed the value of $\chi ^*_{cr}(H)$ and the ‘reason’ it takes this value, then it seems difficult to produce such an explicit extremal example. Instead, the proof splits into a few cases and uses various tools and results that we introduce in the paper.

At this point, the reader may wonder if the conclusion of Theorem 1.6 can be strengthened to ensure a perfect H-tiling. For some ordered graphs H, this is possible. However, for other ordered graphs, one will require a significantly higher minimum degree condition. The following definition is the critical concept for articulating this dichotomy for H with $\chi _<(H) \geq 3$ .

Note that in this definition, we may have that a colour class $V_k$ is empty. If H is the ordered complete graph on r vertices, then H does not have a local barrier; it is also easy to check that $\chi ^*_{cr}(H)=\chi _<(H)=r$ . Given $r \geq 2,$ let $H'$ be any complete r-partite (unordered) graph with at least $2$ vertices in each colour class. Let H be any ordered graph obtained from $H'$ by assigning an interval labelling to $H'$ ; so $\chi _<(H)=r$ . Then one can check that H has a local barrier with parameters $i=1$ and $j=r+1$ as in Definition 1.7.

We are now able to state our main result, which resolves Problem 1.1 for all ordered graphs H with $\chi _<(H) \geq 3$ .

Therefore Problem 1.1 is now asymptotically resolved. The reader might find it hard to see why the value of $\delta _<(H,n)$ behaves as in Theorem 1.8, and indeed, it took the authors quite some time to discover the correct behaviour of this problem. In Section 1.3, we give further intuition on this result. In Section 3, we give examples of H in each case (i)–(iii) of the theorem. In Section 2, we give the extremal constructions for Theorem 1.8. In particular, in cases (i) and (iii), the extremal examples are ‘bottlegraphs’ – complete multipartite ordered graphs where each part has the same size except at most one smaller part; in Section 11, we explicitly compute the value of $\chi ^*_{cr}(H)$ for a range of H and thus the minimum degree threshold in Theorem 1.8 also. The explanation for why these extremal ‘bottlegraphs’ do not have perfect H-tilings revolves around ‘space barrier’ constraints (i.e., one runs out of space in some subset of vertices in the extremal bottlegraph).Footnote ² However, the ‘reason’ for obtaining a space barrier can be somewhat involved (and unlike any other space barrier extremal example we have ever seen before); we discuss this in Section 3.1.

The proof of Theorem 1.8 applies an absorbing theorem from [Reference Balogh, Li and Treglown3, Theorem 4.1] and Theorem 1.6 above. The main novelty is to prove an absorbing theorem for ordered graphs H as in Theorem 1.8(iii). Whilst our argument makes use of a lemma of Lo and Markström [Reference Lo and Markström23] and seems rather natural, it is different to any absorbing proof we have previously seen (in particular, we do not use local-global absorbing as in [Reference Balogh, Li and Treglown3]). See Section 9 for an overview of our absorbing strategy.

1.2 A Dirac-type theorem for vertex covers

Given (ordered) graphs H and G, we say that G has an H-cover if every vertex in G lies in a copy of H. Note that the notion of an H-cover is an ‘intermediate’ between seeking a single copy of H and a perfect H-tiling; in particular, a perfect H-tiling in G is itself an H-cover. Given any $n \in \mathbb N$ and any (ordered) graph H, let $\delta _{\text {cov}}(H,n)$ denote the smallest integer k such that every n-vertex (ordered) graph G with $\delta (G)\geq k$ contains an H-cover. As noted in [Reference Kühn, Osthus and Treglown22], an easy application of Szemerédi’s regularity lemma asymptotically determines $\delta _{\text {cov}}(H,n)$ for all graphs H.

Proposition 1.9 [Reference Kühn, Osthus and Treglown22, Proposition 6]

For every graph H and every $\eta>0$ , there exists an integer $n_0=n_0(H,\eta )$ so that if $n \geq n_0$ , then

$$ \begin{align*} \left (1-\frac{1}{\chi(H)-1} \right )n-1 \leq \delta_{\text{cov}}(H,n)\leq \left (1-\frac{1}{\chi(H)-1} +\eta \right )n. \end{align*} $$

Proposition 1.9 implies that, asymptotically, the minimum degree threshold for ensuring an H-cover in a graph G is the same as the minimum degree threshold for ensuring a single copy of H in G. In [Reference Kühn, Osthus and Treglown22, Theorem 5], Kühn, Osthus and Treglown asymptotically determined the Ore-type degree condition that forces an H-cover for any fixed graph H. There have also been several recent papers concerning minimum $\ell $ -degree conditions that force H-covers in k-uniform hypergraphs; see, for example, [Reference Falgas-Ravry, Markström and Zhao11, Reference Falgas-Ravry and Zhao12, Reference Han, Lo and Sanhueza-Matamala14].

Falgas-Ravry [Reference Falgas-Ravry10] raised the question of determining $\delta _{\text {cov}}(H,n)$ for all ordered graphs H. Our next result asymptotically answers this question.

Theorem 1.10 is a direct consequence of some of the auxiliary results we use in the proof of Theorem 1.8. Note that the behaviour of the threshold in Theorem 1.10 is perhaps unexpected. Indeed, unlike in the unordered setting, Theorem 1.10 and the Erdős–Stone–Simonovits theorem for ordered graphs imply that the asymptotic minimum degree thresholds for forcing a copy of H and an H-cover are different if H has a local barrier.

Furthermore, a key moral of the Erdős–Stone–Simonovits theorem (and Proposition 1.9) is that once a graph G is dense enough (or has large enough minimum degree) to ensure a copy of $K_r$ (or a $K_r$ -cover), then G must contain every fixed graph H (or an H-cover) for every H of chromatic number r. An intuition for this comes from Szemerédi’s regularity lemma. However, the analogous moral is not true for H-covers in ordered graphs. Indeed, if H is an ordered complete graph on r vertices (so $\chi _<(H)=r$ and H has no local barrier), then Theorem 1.10 tells us the minimum degree threshold for forcing an H-cover in an n-vertex ordered graph is $(1-\frac {1}{r-1}+o(1) )n$ whilst the corresponding threshold for any ‘blow-up’ $H'$ of H (so $\chi _<(H')=\chi _<(H)=r$ and H has a local barrier) is significantly higher, namely $(1-\frac {1}{r}+o(1) )n$ . This should hint to the reader that the regularity method behaves differently in the ordered setting; in particular, if H has a local barrier, this provides an obstruction when applying the regularity lemma. More discussion on the regularity method for ordered graphs can be found in [Reference Balogh, Li and Treglown3, Section 3.1].

1.3 Intuition behind the threshold in Theorem 1.8

In this subsection, we build up further intuition behind the threshold in Theorem 1.8. For this, it will be useful to first take a step back and consider perfect H-tilings in unordered graphs. In this setting, the Dirac-type threshold is governed by two factors:

• $(C1)$ The minimum degree needs to be large enough to force an almost perfect H-tiling.

• $(C2)$ The minimum degree must be large enough to prevent ‘divisibility’ barriers within the host graph that constrain us from turning an almost perfect H-tiling into a perfect H-tiling.

This is made precise by the following theorem of Kühn and Osthus [Reference Kühn and Osthus21]. (See [Reference Kühn and Osthus21, Section 1.2] for the definition of $\text {hcf}(H)=1$ .)

Theorem 1.11 (Kühn and Osthus [Reference Kühn and Osthus21])

Let $\delta (H, n)$ denote the smallest integer k such that every graph G whose order n is divisible by $|H|$ and with $\delta (G)\geq k$ contains a perfect H-tiling. For every unordered graph H,

$$\begin{align*}\delta(H, n)=\left(1 - \frac{1}{\chi^*(H)}\right)n +O(1), \end{align*}$$

where $\chi ^*(H):=\chi _{cr} (H)$ if $\text {hcf}(H)=1$ and $\chi ^*(H):=\chi (H)$ otherwise.

Recall that every graph H satisfies $\chi _{cr} (H)\le \chi (H)$ . So by Komlós’ aforementioned almost perfect tiling theorem [Reference Komlós16], the minimum degree condition in Theorem 1.11 is enough to ensure that $(C1)$ holds. Meanwhile, those graphs H with $\text {hcf}(H)=1$ are precisely the graphs for which, at the almost perfect tiling threshold, $(C2)$ is satisfied. Furthermore, for graphs H with $\text {hcf}(H)\ne 1$ , $(C2)$ is only guaranteed to be satisfied once the n-vertex host graph has minimum degree around $(1-1/\chi (H))n$ .

We do not state the precise definition of $\text {hcf}(H)=1$ here; however, the following example is instructive. Let H be any connected bipartite graph, and let $n \in \mathbb N $ be divisible by $|H|$ . Consider the n-vertex graph G that consists of two disjoint cliques whose sizes are as equal as possible so that neither is divisible by $|H|$ . Then whilst G contains an almost perfect H-tiling, the divisibility constraint on the clique sizes prevents a perfect H-tiling. Thus all such H are examples of graphs with $\text {hcf}(H) \ne 1$ . In particular, $\delta (G)=n/2-O(1)=(1-1/\chi (H))n-O(1)$ , so G is an extremal example for Theorem 1.11 in this case.

As mentioned earlier, another necessary condition for a Dirac-type threshold for perfect H-tilings is the following:

• $(C3)$ The minimum degree needs to be large enough to force an H-cover.

Condition $(C3)$ , however, does not factor into the statement of Theorem 1.11 as Proposition 1.9 shows that one can ensure an H-cover ‘earlier’ than an almost perfect H-tiling (recall that $\chi (H)-1< \chi _{cr}(H)$ ).

Interestingly, the opposite is true in the ordered graph setting when H has a local barrier and $\chi _<(H)\geq 3$ is such that $\chi _<(H)> \chi _{cr}^* (H)$ . Indeed, in this case, Theorem 1.8(ii) essentially states that the H-cover condition is the ‘last’ of conditions $(C1)$ – $(C3)$ to be satisfied. In particular, Extremal Example 1 in Section 2 shows that for every H satisfying Definition 1.7, there are n-vertex ordered graphs with $\delta (G)> (1-1/\chi _<(H))n-1$ for which a certain vertex does not lie in a copy of H.

In all other cases when $\chi _<(H)\geq 3$ , Theorem 1.8 essentially states that the almost perfect tiling condition is the ‘last’ of conditions $(C1)$ – $(C3)$ to be satisfied. Therefore, surprisingly (at least to the authors!), divisibility barriers play no role in Problem 1.1 for H with $\chi _<(H)\geq 3$ . In contrast, Theorem 1.9 in [Reference Balogh, Li and Treglown3] shows that in the case when $\chi _<(H)=2$ , each of $(C1)$ , $(C2)$ and $(C3)$ can be the condition that governs the Dirac-type threshold for perfect H-tiling, depending on the choice of H.

1.4 A Dirac-type theorem for H-tilings

In addition to determining the Dirac-type threshold for almost perfect tilings in unordered graphs, Komlós [Reference Komlós16] provided a best-possible minimum degree condition for forcing an H-tiling covering a certain proportion of the vertices in a graph G.

Theorem 1.13 (Komlós [Reference Komlós16])

Let H be a graph and $x\in (0,1)$ . Define

$$ \begin{align*} g(x,H):=\left(1-\frac{1}{\chi(H)-1}\right)(1-x)+\left(1-\frac{1}{\chi_{cr}(H)}\right)x. \end{align*} $$

Given any $\eta>0$ , there exists some $n_0=n_0(x,H,\eta )\in \mathbb {N}$ such that if G is a graph on n vertices where $n\geq n_0$ and $\delta (G)\geq g(x,H)\cdot n$ , then there exists an $(x-\eta ,H)$ -tiling in G.

Note that the minimum degree condition in Theorem 1.13 is the best possible in the sense that given any fixed H and x ∈ (0, 1), one cannot replace g(x, H) with any fixed g′(x, H) < g(x, H) (see [Reference Komlós16, Theorem 7] for a proof of this).

The function $g(x,H)$ is quite well-behaved. Indeed, for fixed H, $g(x,H)$ grows linearly in x. Note that $g(0,H)\cdot n$ and $g(1,H)\cdot n$ are the asymptotic minimum degree thresholds for ensuring an n-vertex graph contains a copy of H and an almost perfect H-tiling, respectively. From this perspective, the function $g(x,H)$ can be viewed as a linear interpolation of these two thresholds.

The question of obtaining an ordered graph analogue of Theorem 1.13 was raised in [Reference Balogh, Li and Treglown3, Question 8.2]. We provide an answer to this problem; for this, we require the following definitions.

Definition 1.15. Let H be an ordered graph and $x\in (0,1]$ . We define $\chi _{cr}^*(x,H)$ as

$$ \begin{align*} \chi_{cr}^*(x,H):=\inf \{\chi_{cr}(B) \, : \, B \, \text{is an } x\text{-bottlegraph of } H\}. \end{align*} $$

Given any ordered graph H, if B is an x-bottlegraph of H, then $\chi (B) \geq \chi _<(H)$ . This implies that $\chi _{cr}(B)> \chi _<(H)-1$ , so

(1) $$ \begin{align} \chi ^* _{cr}(x,H) \geq \chi _< (H)-1. \end{align} $$

An application of Theorem 1.13 together with a tool from [Reference Balogh, Li and Treglown3, Lemma 6.2] yields the following minimum degree condition for the existence of $(x,H)$ -tilings in ordered graphs.

Theorem 1.16. Let H be an ordered graph and $x\in (0,1)$ , and define

$$ \begin{align*} f(x,H):=\left(1-\frac{1}{\chi_{cr}^*(x,H)}\right). \end{align*} $$

Given any $\eta>0$ , there exists some $n_0=n_0(x,H,\eta )\in \mathbb {N}$ such that if G is an ordered graph on n vertices with $n\geq n_0$ and $\delta (G)\geq (f(x,H)+\eta ) n$ , then G contains an $(x,H)$ -tiling.

The minimum degree condition in Theorem 1.16 is the best possible in the following sense. Let H and $x\in (0,1)$ be fixed. Given any $0<a <1-1/\chi _{cr}^*(x,H)$ and any sufficiently large $n \in \mathbb N$ , consider any n-vertex graph B that satisfies (i) and (ii) of Definition 1.14 for some choice of $k,m \in \mathbb N$ and where

$$ \begin{align*} an < \delta (B) = n-m= \left(1-\frac{1}{\chi_{cr}(B)}\right)n< f(x,H)\cdot n. \end{align*} $$

So $\chi _{cr}(B) <\chi _{cr}^*(x,H)$ . (Note such a graph B exists for any choice of $0<a <1-1/\chi _{cr}^*(x,H)$ .) Then by definition of $\chi _{cr}^*(x,H)$ , there is a permutation $\sigma $ of $[k]$ and an interval labelling $\phi $ of B with respect to $\sigma $ such the resulting ordered graph $(B,\phi )$ does not contain an $(x,H)$ -tiling.

A drawback of Theorem 1.16 is that it seems hard to compute $\chi _{cr}^*(x,H)$ in general. However, in Section 14, we describe the behaviour of the function $f(x,H)$ for some fixed ordered graphs H. In particular, akin to Theorem 1.13, if H has $\chi _<(H)=2$ , then $f(x,H)$ is linear in x. Perhaps surprisingly, though, there are ordered graphs where $f(x,H)$ is only piecewise linear. We also compute $f(x,H)$ for every ordered graph H and every x that is not too big.

1.5 Organisation of the paper

The paper is organised as follows. In Section 2, we give the extremal constructions for Theorems 1.8 and 1.10. In Section 3, we give some examples of ordered graphs H that fall into each of the three cases of Theorem 1.8. In Section 4, we state a new absorbing theorem (Theorem 4.2) and an absorbing theorem from [Reference Balogh, Li and Treglown3] and combine them with Theorem 1.6 to prove Theorem 1.8. The subsequent sections therefore build up tools for the proof of Theorem 4.2: in Section 5, we state a couple of useful properties of bottlegraphs; in Section 6, we introduce Szemerédi’s regularity lemma and related useful results; some tools for absorbing are given in Section 7; Section 8 contains several results that give flexibility in how one can interval colour certain ordered graphs H.

In Section 9, we give a sketch of the proof of Theorem 4.2 before proving it and Theorem 1.10 in Section 10. In Section 11, we give general upper and lower bounds on $\chi ^*_{cr}(H)$ and also compute $\chi ^*_{cr}(H)$ for a few general classes of ordered graphs H.

In Section 12, we prove that the minimum degree condition in Theorem 1.6 is the best possible. The proof of Theorem 1.16 is given in Section 13; in the subsequent section, we describe the behaviour of the function $f(x,H)$ for some choices of H. We conclude the paper with some open problems in Section 15.

1.6 Notation

Given $n\in \mathbb N$ , let $[n]:=\{1, \ldots , n\}$ . A nearly balanced interval partition of $[n]$ is a partition of $[n]$ into intervals $W_1<\dots <W_t$ , where $||W_i|-|W_j||\leq 1$ for every $1\leq i, j \leq t$ . Similarly, a t-partite graph with vertex classes $V_1,\ldots , V_t$ is nearly balanced if $||V_i|-|V_j||\leq 1$ for every $1\leq i, j \leq t$ .

If G is an (ordered) graph, $|G|$ denotes the size of its vertex set, and $e(G)$ denotes the number of edges in G. Given $A\subseteq V (G)$ , the induced subgraph $G[A]$ is the subgraph of G whose vertex set is A and whose edge set consists of all of the edges of G with both endpoints in A. We define $G\setminus A:=G[V(G)\setminus A]$ . For two disjoint subsets $A, B\subseteq V (G)$ , the induced bipartite subgraph $G[A, B]$ is the subgraph of G whose vertex set is $A\cup B$ and whose edge set consists of all of the edges of G with one endpoint in A and the other endpoint in B. We write $e(A,B):=e(G[A,B])$ .

For an (ordered) graph G and a vertex $x \in V(G)$ , we define $N_G(x)$ as the set of neighbours of x in G and $d_G (x):=|N_G(x) |$ . For $X \subseteq V(G)$ , we define $d_G (x,X):=|N_G(x)\cap X |$ . Given (ordered) graphs G and H and $X \subseteq V(G)$ , we say that $G[X]$ spans a copy of H in G if H is a spanning subgraph of $G[X]$ .

Given an ordered graph G, we say that $V_1<\dots <V_r$ is an interval r-colouring of G to mean there is an interval r-colouring of G with colour classes $V_1< \dots <V_r$ . We say that an ordered graph G is complete r-partite if there exists an interval r-colouring $V_1<\cdots <V_r$ such that $xy\in E(G)$ for every $x\in V_i$ and $y\in V_j$ with $i\not =j$ . We refer to the $V_i$ s as the parts of G.

Given an unordered graph G and a positive integer t, let $G(t)$ be the graph obtained from G by replacing every vertex $x\in V(G)$ by a set $V_x$ of t vertices spanning an independent set and joining $u\in V_x$ to $v\in V_y$ precisely when $xy$ is an edge in G; that is, we replace the edges of G by copies of $K_{t,t}$ . We will refer to $G(t)$ as a blown-up copy of G. If $U_i$ is a vertex class in G, then we write $U_i(t)$ for the corresponding vertex class in $G(t)$ . We use analogous notation when considering blown-up copies of complete k-partite ordered graphs. In particular, given a complete k-partite ordered graph B with parts $B_1<\dots < B_k$ , the ordered blow-up $B(t)$ of B consists of parts $B_1(t)<\dots < B_k(t)$ , where $|B_i(t)|=t|B_i|$ for all $i \in [k]$ .

Throughout the paper, we omit all floor and ceiling signs whenever they are not crucial. The constants in the hierarchies used to state our results are chosen from right to left. For example, if we claim that a result holds whenever $0< a\ll b\ll c\le 1$ , then there are nondecreasing functions $f:(0,1]\to (0,1]$ and $g:(0,1]\to (0,1]$ such that the result holds for all $0<a,b,c\le 1$ with $b\le f(c)$ and $a\le g(b)$ . Note that $a \ll b$ implies that we may assume in the proof that, for example, $a < b$ or $a < b^2$ .