1.1 An optimisation problem

This paper concerns the relation between the structure of almost extremal graphs for $F(n,\boldsymbol {k})$ and optimal solutions of a certain optimisation problem, Problem $Q_t$ , which we now define.

Problem $Q_t$ : Given a sequence $\boldsymbol {k} := (k_1,\ldots ,k_s) \in \mathbb {N}^s$ of natural numbers and $t\in \{0,1,2\}$ , determine

(1.3)

the maximum value of

(1.4) $$ \begin{align} q(\phi,\boldsymbol{\alpha}) := 2\sum_{\stackrel{1 \leq i < j \leq r}{\phi(ij) \neq \emptyset}} \alpha_i \alpha_j \log_2 |\phi(ij)| \end{align} $$

over the set of feasible solutions, that is, triples $(r,\phi ,\boldsymbol {\alpha })$ , such that

1. • $r \in \mathbb {N}$ and $\phi \in \Phi _t(r;\boldsymbol {k})$ , where $\Phi _t(r;\boldsymbol {k})$ is the set of all functions $\phi : \binom {[r]}{2} \rightarrow 2^{[s]}$ , such that

   $$\begin{align*}\phi^{-1}(c) := \left\{ ij \in \binom{[r]}{2} : c \in \phi(ij) \right\} \end{align*}$$
   is $K_{k_c}$ -free for every colour $c \in [s]$ and $|\phi (ij)|\ge t$ for all $ij\in \binom {[r]}{2}$ ;

2. • $\boldsymbol {\alpha } = (\alpha _1,\ldots ,\alpha _r) \in \Delta ^r$ , where $\Delta ^r$ is the set of all $\boldsymbol {\alpha } \in \mathbb {R}^r$ with $\alpha _i \geq 0$ for all $i \in [r]$ , and $\alpha _1 + \ldots + \alpha _r = 1$ .

Note that for $t=1,2$ , a triple necessarily has $r<R(\boldsymbol {k})$ where $R(\boldsymbol {k})$ is the Ramsey number of $\boldsymbol {k}$ (i.e. the minimum R, such that $K_R$ admits no $\boldsymbol {k}$ -valid s-edge-colouring). Thus, the maximum in (1.3) is attained for $t=1,2$ since $q(r,\phi ,\cdot )$ is continuous for each of the finitely many pairs $(r,\phi )$ , and is a (nonempty) compact space. It is also attained for $t=0$ by (1.5) below. We call $\phi \in \Phi _0(r;\boldsymbol {k})$ a colour pattern and $\boldsymbol {\alpha } \in \Delta ^r$ a vertex weighting. A triple $(r,\phi ,\boldsymbol {\alpha })$ is called $Q_t$ -optimal if it attains the maximum, that is, and $q(r,\phi ,\boldsymbol {\alpha })=Q_t(\boldsymbol {k})$ . One can easily show [Reference Pikhurko, Staden and Yilma24, Lemma 6] that

(1.5) $$ \begin{align} Q(\boldsymbol{k}) := Q_2(\boldsymbol{k}) = Q_1(\boldsymbol{k})=Q_0(\boldsymbol{k}). \end{align} $$

Note that a $Q_0$ -optimal triple can have r arbitrarily large, by, for example, adding vertices of weight 0 or splitting an existing vertex into two clones. Let be the set of $Q_t$ -optimal triples $(r,\phi ,\boldsymbol {\alpha })$ . Let be the set of with $\alpha _i>0$ for every $i \in [r]$ . Let be the set of basic optimal solutions, which are $Q_2$ -optimal triples $(r,\phi ,\boldsymbol {\alpha })$ with $\alpha _i>0$ for every $i \in [r]$ .

Given vectors $\boldsymbol {a}=(a_1,\ldots ,a_s)$ and $\boldsymbol {b}=(b_1,\ldots ,b_t)$ , write $\boldsymbol {a}\leq \boldsymbol {b}$ if $a_i \leq b_i$ for all $i \leq \max \{s,t\}$ where $a_i:=0$ for all $i>s$ and $b_i:=0$ for all $i>t$ . We write $\| \boldsymbol {a}-\boldsymbol {b}\|_1 := \sum _{i \leq \max \{s,t\}}|a_i-b_i|$ for the $\ell ^1$ -distance between $\boldsymbol {a}$ and $\boldsymbol {b}$ . In this paper, we always take $\log $ to the base $2$ ; from now on, we omit any subscript.

One should think of feasible triples $(r,\phi ,\boldsymbol {\alpha })$ as vertex-weighted edge-coloured multigraphs. It is not hard to show that $F(\boldsymbol {k})\geq Q(\boldsymbol {k})$ . Indeed, given $\boldsymbol {k}$ , a $Q_1$ -optimal triple $(r,\phi ,\boldsymbol {\alpha })$ and $n \in \mathbb {N}$ , take the complete r-partite n-vertex graph $K_{\boldsymbol {\alpha }}(n)$ whose parts $X_1,\ldots ,X_r$ satisfy $|\,|X_i|-\alpha _i n\,| \leq 1$ for all $i \in [r]$ . Consider those s-edge-colourings of $K_{\boldsymbol {\alpha }}(n)$ in which, for $x \in X_i$ and $y \in X_j$ , we only allow colours in $\phi (ij)$ to be used on $xy$ . Every such colouring is $\boldsymbol {k}$ -valid, since every $\phi ^{-1}(c)$ is $K_{k_c}$ -free. Clearly, $F(n;\boldsymbol {k})$ is bounded below by the number of such colourings of $K_{\boldsymbol {\alpha }}(n)$ , which is

$$ \begin{align*}\prod_{ij\in\binom{[r]}{2}}|\phi(ij)|^{|X_i||X_j|} = 2^{q(\phi,\boldsymbol{\alpha})n^2/2+O(n)} = 2^{Q(\boldsymbol{k})n^2/2+O(n)}. \end{align*} $$

Taking the limit as $n\to \infty $ , we have $F(\boldsymbol {k})\geq Q(\boldsymbol {k})$ . With Yilma, we proved the following results relating the determination of $F(n;\boldsymbol {k})$ to Problem $Q_1$ , including a matching upper bound.

Thus, by (1.5), to determine $F(\boldsymbol {k})$ , it suffices to find the optimal solutions to Problem $Q_2$ , which has the smallest feasible set among the problems $Q_t$ . Unfortunately, this is difficult even when $\boldsymbol {k}$ is small. Given a pair $(r,\phi )$ , one can use the method of Lagrange multipliers to find a best possible $\boldsymbol {\alpha }$ for this pair; though the number of pairs $(r,\phi )$ is finite, there are generally too many for a computer search. Indeed, the upper bound of $R(\boldsymbol {k})$ for r grows large very quickly, though we expect the optimal r to be much smaller than $R(\boldsymbol {k})$ .

1.2 New results

The main contribution of this paper is a general stability theorem that determines the structure of any n-vertex graph G which is almost $\boldsymbol {k}$ -extremal, that is with $F(G,\boldsymbol {k})=F(n;\boldsymbol {k})\cdot 2^{o(n^2)}$ . This will show that the structure of any such graph is similar to an optimal solution to Problem $Q_0$ , and almost all valid colourings almost follow an optimal colour pattern. This stability result holds for all $\boldsymbol {k}$ satisfying a rather general condition, which we call the extension property. Given , one can easily check whether this condition holds. Indeed, we show that in almost all instances for which $F(\boldsymbol {k})$ is known, $\boldsymbol {k}$ satisfies a strong version of this property.

We explain the intuition and motivation behind this property in Section 2.1. For now, we remark that it is generally easy to check whether $\boldsymbol {k}$ has the extension property, when is known. We check it for some cases in Section 5. For $\boldsymbol {k}$ with the extension property, one can describe all solutions to in terms of basic optimal solutions. Every solution can be obtained by ‘blowing up’ a basic optimal solution $(r^*,\phi ^*,\boldsymbol {\alpha }^*)$ ; that is, taking arbitrarily many clones of the vertices and modifying part sizes so that the sum of vertex weights of clones of j equals $\alpha _j^*$ for every j; and then possibly adding colour c edges between clones of each j, without creating a c-coloured copy of $K_{k_c}$ , where c is the colour with the largest forbidden clique. Without loss of generality, we assume $k_1 \geq \ldots \geq k_s$ , so that $c=1$ . If $k_1=k_2$ , then one cannot add any colour $1$ edges between clones without creating a forbidden clique.

Our main result is the following stability theorem. The edit distance $d_{\mathrm {edit}}(G,G')$ of two graphs $G,G'$ of the same order is the minimum number of edges that need to be added/removed to make $G'$ isomorphic to G. Given graphs G and H and $\delta>0$ , we say that G is $\delta $ -far from being H-free if it has edit distance at least $\delta |V(G)|^2$ to every H-free graph with the same number of vertices (note that we only need to delete edges here). Given disjoint $A,B \subseteq V(G)$ and $0 \leq d \leq 1$ , we say that $G[A,B]$ is $(\delta ,d)$ -regular if $d_G(A,B) := e_G(A,B)|A|^{-1}|B|^{-1} \in (d-\delta ,d+\delta )$ , and $|d_G(X,Y)-d_G(A,B)|<\delta $ for all $X\subseteq A$ , $Y \subseteq B$ with $|X|/|A|, |Y|/|B| \geq \delta $ . We are now ready to state the stability theorem. It says that, for any large n-vertex graph G which has close to the maximum number of valid colourings; that is, $F(G;\boldsymbol {k})=F(n,\boldsymbol {k}) \cdot 2^{o(n^2)}$ , for almost all of its valid colourings $\chi $ , there is a solution to which describes the structure of $\chi $ : it looks like a ‘blow-up’ of the solution. Lemma 1.3 describes the structure of solutions in terms of basic optimal solutions, and, therefore, there is a basic optimal solution $(r^*,\phi ^*,\boldsymbol {\alpha }^*)$ which describes the structure of $\chi $ . Part (ii) implies that, not only is there a partition of $V(G)$ , such that $\chi $ has many edges of every colour $c \in \phi ^*(ij)$ between the i-th and j-th parts (and few edges of any other colour), these edges are in fact well-distributed and of roughly equal densities between these parts.

Theorem 1.4 (Stability).

Let $s \in \mathbb {N}$ , and suppose that $\boldsymbol {k} \in \mathbb {N}^s$ with $k_1\geq \ldots \geq k_s$ has the extension property. Then for all $\delta> 0$ , there exist $n_0 \in \mathbb {N}$ and $\varepsilon> 0$ , such that the following holds. If G is a graph on $n \geq n_0$ vertices, such that

$$ \begin{align*}\frac{\log F(G;\textbf{k})}{\binom{n}{2}} \geq Q(\textbf{k}) - \varepsilon, \end{align*} $$

then, for at least $(1-2^{-\varepsilon n^2})F(G;\boldsymbol {k})$ colourings $\chi : E(G) \rightarrow [s]$ which are $\boldsymbol {k}$ -valid, there are and a partition $Y_1 \cup \ldots \cup Y_{r^*}$ of $V(G)$ , such that the following hold.

1. (i) For all $i \in [r^*]$ , we have that $|\,|Y_i| - \alpha ^*_in\,| < 1$ .

2. (ii) For all $c \in \phi ^*(ij)$ and $ij \in \binom {[r^*]}{2}$ , we have that $\chi ^{-1}(c)[Y_i,Y_j]$ is $(\delta ,|\phi ^*(ij)|^{-1})$ -regular. In particular, $e_G(Y_i,Y_j) \geq (1-s\delta )|Y_i||Y_j|$ .

3. (iii) Suppose $\sum _{i \in [r^*]}e(G[Y_i])> \delta n^2$ . Then $\boldsymbol {k}$ does not have the strong extension property, and all but at most $\delta n^2$ edges in $\bigcup _{i \in [r^*]}G[Y_i]$ are coloured with $1$ under $\chi $ . Moreover, if $\boldsymbol {\ell } := (\ell _1,\ldots ,\ell _{r^*}) \in \mathbb {N}^{r^*}$ is such that $G[Y_i]$ is $\delta $ -far from being $K_{\ell _i}$ -free, then $\|\boldsymbol {\ell }\|_1 \leq k_{1}-1$ .

Somewhat conversely, if (i), (iii) and $e_G(Y_i,Y_j)\geq (1-s\delta )|Y_i||Y_j|$ hold for some triple , a partition $Y_1,\ldots ,Y_{r^*}$ of an n-vertex graph G, and $\delta =o(1)$ , and each $G[Y_i]$ is $K_{\ell_i+1}$ -free for some vector $\boldsymbol {\ell }$ of $1$ -norm at most $k_1-1$ , then $F(G;\boldsymbol {k})\geq 2^{(Q(\boldsymbol {k})-o(1))n^2/2}$ .

One should note the similarities between the statements of Theorem 1.4 and Lemma 1.3. Indeed, this parallel shows that the gist of Theorem 1.4 is ‘near-extremal graphs look like blow-ups of solutions to ’. This is not quite true within parts, as here, G could be very far from a complete partite graph. Note also that the partition $Y_1 \cup \ldots \cup Y_{r^*}$ may be different for different colourings $\chi $ .

We illustrate these statements with the example $\boldsymbol {k} = (5,3)$ . As it is not hard to show (or see Lemma 1.8), consists of just one element, namely, $(2,\phi ^*,(\frac {1}{2},\frac {1}{2}))$ , where $\phi ^*(12):=\{1,2\}$ . Thus, by Theorem 1.4 and the remark after it, the set of almost extremal graphs can be described as consisting of graphs that are $o(n^2)$ -close to $T_2(n)$ with triangle-free graphs added into each part, or a $K_4$ -free graph added into one part. Note that the partition $Y_1\cup Y_2$ of Theorem 1.4 may depend on the colouring. For example, if G is $T_4(n)$ with parts $V_1,\ldots ,V_4$ , then Theorem 1.4 gives that for a typical colouring, there are disjoint pairs $ij,\ell m$ , such that almost all edges between $V_i$ and $V_j$ and between $V_{\ell }$ and $V_m$ are coloured with colour $1$ ; then $Y_1$ and $Y_2$ in Theorem 1.4 have to be $V_i\cup V_j$ and $V_{\ell } \cup V_m$ , up to changing $o(n)$ vertices.

If $\boldsymbol {k}$ has the strong extension property, then G is close to a complete multipartite graph by Theorem 1.4. So the pairs $(r^*,\boldsymbol {\alpha }^*)$ associated with the colourings specified by the theorem are close to each other up to a relabelling of colours. We have the following corollary of Theorem 1.4 for $\boldsymbol {k}$ with the strong extension property.

Corollary 1.5. Let $s \in \mathbb {N}$ , and suppose that $\boldsymbol {k} \in \mathbb {N}^s$ with $k_1\geq \ldots \geq k_s$ has the strong extension property. Then for all $\delta> 0$ , there exists $n_0 \in \mathbb {N}$ and $\varepsilon> 0$ , such that the following holds. Let G be a graph on $n \geq n_0$ vertices, such that

$$ \begin{align*} \frac{\log F(G;\boldsymbol{k})}{\binom{n}{2}} \geq Q(\boldsymbol{k}) - \varepsilon. \end{align*} $$

Then there are $r^*,\boldsymbol {\alpha }^*$ and a partition V(G) = $V_1 \cup \ldots \cup V_{r^*}$ with $\left |\,|V_i|-\alpha ^*_i n \,\right | < 1$ for all $i \in [r^*]$ , such that the edit distance between G and $K[V_1,\ldots ,V_{r^*}]$ is at most $\delta n^2$ . Moreover, for at least $(1-2^{-\varepsilon n^2}) \cdot F(G;\boldsymbol {k}) \ \boldsymbol {k}$ -valid s-edge-colourings $\chi $ of G, there exists where $\|\boldsymbol {\alpha }-\boldsymbol {\alpha }^*\|_1 \leq \delta $ , such that $\chi ^{-1}(c)[V_i,V_j]$ is $(\delta ,|\phi ^*(ij)|^{-1})$ -regular for all $ij \in \binom {[r^*]}{2}$ and $c \in \phi ^*(ij)$ .

Recall from Theorem 1.1 (i) that at least one extremal graph is complete multipartite. The following conjecture was made in [Reference Pikhurko, Staden and Yilma24]:

In [Reference Pikhurko and Staden23], we will use Corollary 1.5 to prove an exact result for all $\boldsymbol {k}$ with the strong extension property: that for such $\boldsymbol {k}$ , every large extremal graph is a complete multipartite graph with part ratios roughly $\alpha _1^*,\ldots ,\alpha _{r^*}^*$ for some $\boldsymbol {\alpha }^*$ coming from a basic optimal triple $(r^*,\phi ^*,\boldsymbol {\alpha }^*)$ . However, it seems much harder to prove an exact result for $\boldsymbol {k}$ without the strong extension property, as Theorem 1.4 as well as the example $\boldsymbol {k}=(5,3)$ above show that, in general, there are many asymptotically extremal graphs which are far from complete multipartite.

1.3 Applications

Armed with Theorem 1.4, to determine asymptotically $\boldsymbol {k}$ -extremal graphs, one need only solve Problem $Q_2$ for $\boldsymbol {k}$ , and then check for the extension property using the optimal solutions.

We apply our stability theorem to reprove stability for most of the cases in which $F(n;\boldsymbol {k})$ has already been determined, in a systematic fashion. For this, it suffices to solve Problem $Q_2$ (which follows from these earlier works), and to prove the extension property. Proving the extension property is straightforward: there are $O(s^r)$ possible attachments of a new vertex to some $(r,\phi ,\boldsymbol {\alpha })$ so a computer could check these for reasonable $s,r$ . Actually, when optimal solutions have a particular nice form, which they do in almost all known cases, one can reduce the problem to determining solutions to some simple exponential equation over the integers (see Lemma 5.3). We cannot prove stability for $\boldsymbol {k}=(3,3,3,3,3)$ , which does not have the extension property, strong or otherwise.

As an example, we solve the optimisation problem for $s=2$ and state a version of Theorem 1.4 to illustrate it. Note that it would not be too difficult to prove stability for $s=2$ directly.

We write $\omega (G)$ for the clique number of a graph G; that is, the size of its largest clique.

1.4 A sketch of the proof of Theorem 1.4

Since the proof of Theorem 1.4 is quite involved, we give a fairly detailed sketch first. Let G be a graph on n vertices, such that $\log F(G;\boldsymbol {k}) \geq (Q(\boldsymbol {k})-\varepsilon ) \binom {n}{2}$ . There are several ingredients to the proof.

Regularity lemma: G is close to some nearly optimal $(r,\phi ,\boldsymbol {\alpha })$ in . The multicolour version of Szemerédi’s regularity lemma was already used to prove the existing results on $F(n;\boldsymbol {k})$ in [Reference Alon, Balogh, Keevash and Sudakov1, Reference Pikhurko and Yilma25] (for definitions and statements related to the regularity lemma, see Section 4.1). Given a $\boldsymbol {k}$ -valid colouring $\chi $ of G, we obtain an equitable partition $U_1 \cup \ldots \cup U_r$ in which almost all pairs are regular in all colours in $[s]$ . Define a colour pattern $\phi : \binom {[r]}{2} \rightarrow 2^{[s]}$ by adding the colour c to $\phi (ij)$ if $\chi ^{-1}(c)[V_i,V_j]$ is regular, and has density that is not too small. The embedding lemma (Lemma 4.6) implies that $\phi ^{-1}(c)$ is $K_{k_c}$ -free for all $c \in [s]$ . In this way, much of the information carried by $\chi $ is transferred to the tuple $\mathop {\mathrm {RL}}(\chi ) := (r,\phi ,\mathcal {U})$ , where $\mathcal {U} := \{ U_1,\ldots ,U_r \}$ .

Of course, one still needs to prove that this process is in some sense reversible: that the structure of G itself, as well as the structure of its colourings, can be recovered from $(r,\phi ,\mathcal {U})$ . This may not always be the case: we could have chosen some pathological $\chi $ to generate $(r,\phi ,\mathcal {U})$ . For example, in the case $\boldsymbol {k} = (3,3)$ , the unique extremal graph is $T_2(n) = K_{\lfloor n/2 \rfloor , \lceil n/2 \rceil }$ , but we could have chosen $\chi $ which colours every edge with colour $1$ . Then we cannot recover many further colourings from $(r,\phi ,\mathcal {U})$ .

For this reason, we only wish to consider tuples $(r,\phi ,\mathcal {U})$ which are the image of many colourings; that is, some nontrivial proportion of all colourings. Such a tuple is called popular; and we think of colourings $\chi $ which map to this tuple as being good representatives of the set of all colourings of G. Since, as we show in Proposition 4.10, almost every colouring maps to a popular tuple, it suffices to fix a popular tuple $(r,\phi ,\mathcal {U})$ and only consider colourings which map to this tuple. Intuitively, all such colourings should be similar.

Let $\alpha _i := 1/r$ for all $i \in [r]$ . Then . So the regularity lemma allows us to pass from G to a feasible solution to Problem $Q_0$ . It turns out that since $(r,\phi ,\mathcal {U})$ is popular, we have that

$$ \begin{align*} q(\phi,\boldsymbol{\alpha}) \geq Q(\boldsymbol{k}) - 2\varepsilon,\\[-15pt] \end{align*} $$

and, moreover, that $G[U_i,U_j]$ is almost complete for all $ij \in \binom {[r]}{2}$ (see Claim 4.1). Since we can choose r large (but still bounded), the number of edges of G within any $U_i$ can be made very small. Therefore, the structure of G can be recovered from $(r,\phi ,\mathcal {U})$ .

Symmetrisation: from to some . This is the main part of the proof (Lemma 3.1), and in it, we forget about G entirely, and instead concentrate on $(r,\phi ,\boldsymbol {\alpha })$ . We think of this object as a vertex-weighted coloured multigraph: the weights are given by $\boldsymbol {\alpha }$ , and the coloured edges by $\phi $ . We will apply a version of symmetrisation to $(r,\phi ,\boldsymbol {\alpha })$ . Symmetrisation was originally used in (ordinary) graphs by Zykov [Reference Zykov30] to give a new proof of Turán’s theorem. In its most basic form, it is the process of considering two nonadjacent vertices x and y in a graph G, and replacing x by a clone of y, that is a vertex $y'$ whose neighbourhood is the same as that of y. With Yilma [Reference Pikhurko, Staden and Yilma24], we used symmetrisation to modify any $\boldsymbol {k}$ -extremal graph into one which is both extremal and complete multipartite (Theorem 1.1 (i)). Here, we use a version of symmetrisation as follows. Suppose that there is some $ij \in \binom {[r]}{2}$ , such that $|\phi (ij)| \leq 1$ . Then we create a new feasible solution on r parts by making vertex j a clone of vertex i, or vice versa. One of these choices will be such that the new solution $(r,\phi ',\boldsymbol {\alpha })$ satisfies $q(\phi ',\boldsymbol {\alpha }) \geq q(\phi ,\boldsymbol {\alpha })$ . At the end of this process, we will obtain (where f is for final), such that $|\phi _f(ij)| \geq 2$ whenever $i,j$ are not clones and

$$ \begin{align*}q(\phi_f,\boldsymbol{\alpha}) \geq Q(\boldsymbol{k}) - 2\varepsilon\\[-15pt] \end{align*} $$

(in fact, we split each step into small steps to get slowly evolving colour patterns $\phi =\phi _0,\ldots ,\phi _f$ ). The solution $(r,\phi _f,\boldsymbol {\alpha })$ corresponds to a smaller solution $(r_f,\psi _f,\boldsymbol {\alpha }_f)$ in which all clones are merged, so (and $q(\psi _f,\boldsymbol {\alpha }_f)=q(\phi _f,\boldsymbol {\alpha })$ is near-optimal). A compactness argument (Lemma 2.3) shows that there is some basic optimal solution $(r^*,\phi ^*,\boldsymbol {\alpha } ^*)$ which is very close to the near-optimal $(r_f,\phi _f,\boldsymbol {\alpha }_f)$ in a very strong sense: $\boldsymbol {\alpha }^*$ and $\boldsymbol {\alpha }_f$ are close in $\ell ^1$ -distance, and $\phi _f$ equals $\phi ^*$ between pairs with nonnegligible weights.

The extension property: $(r,\phi,\boldsymbol{\alpha})$ and $(r^*,\phi^*,\boldsymbol{\alpha}^*)$ are close. Here, we mean ‘close’ in the sense of Lemma 1.3. So we would like to show that when we merge pairs $ij$ with $|\phi(ij)| \leq 1$ in $(r,\phi,\boldsymbol{\alpha})$ we obtain a weight vector $\boldsymbol {u}$ which is close to $\boldsymbol{\alpha}^*$ in $\ell^1$ -distance, and $\phi \subseteq \phi^*$ on pairs of non-negligible size. (This turns out to be a simplification; see below.) It is of course far from clear that we have not changed $(r,\phi,\boldsymbol{\alpha})$ drastically to obtain $(r^*,\phi^*,\boldsymbol{\alpha}^*)$ . That this is not so is effectively a consequence of the extension property.

Having obtained $(r^*,\phi^*,\boldsymbol{\alpha}^*)$ from symmetrisation, we can now, with the benefit of hindsight, follow the procedure backwards. At each stage, we check whether the attachments of each vertex are large; if not we sequentially put bad vertices into an exceptional set, called $U_{i}^0$ at the i-th step. At each stage, the extension property implies that every vertex x is either in $U_{i}^0$ or it corresponds to some vertex k in the optimal solution $(r^*,\phi^*,\boldsymbol{\alpha}^*)$ ; that is, $\phi_i(xy) \subseteq \phi^*(kj)$ for all $j \in [r^*] \setminus \{ k \}$ and all vertices y corresponding to j. Let $U_{i}^k$ be the set of vertices corresponding to k at Step i. Note that $|\phi_i(xy)| \leq 1$ if $x,y \in U_{i}^k$ .

So, going back through the procedure, at each step there are a small fraction of vertices x which were exceptional but are no longer exceptional, and these are moved to the $U_{i}^k$ for which x corresponds to k; and there are a small fraction of vertices moved into $U_{i}^0$ . When we return to Step 0, we want to show that $\phi \subseteq \phi^*$ between any two classes $U_0^j$ and $U_0^k$ , even though the attachment between $U_{0}^0$ and the rest of the vertices, as well as the colour pattern within a class $U_{i}^k$ may have changed. Therefore, if we can show that $U_{0}^0$ is small (Claim 3.1.2) and every $U_{0}^k$ , $k>0$ , is about the same size throughout, then the procedure did not really change $(r,\phi,\boldsymbol{\alpha})$ much at all. Furthermore, we have a partition $U_{0}^0,U_{0}^1,\ldots,U_{0}^{r^*}$ of $[r]$ such that $\phi \subseteq \phi^*$ between $U_{0}^i,U_{0}^j$ , $i,j \neq 0$ ; and $|\phi(ij)| \leq 1$ within a class. In fact, the above sketch is a simplification, and though the exceptional set is always small, the other parts could change significantly in size. Nevertheless, we show that for each i there are $\tilde{r}_i$ nonzero nonexceptional parts with part ratios given by some $\tilde{\boldsymbol{\alpha}}_i$ , and , where $\phi_i\subseteq\tilde{\phi}_i$ between pairs and $|\phi_i(jk)| \leq 1$ within a class.

Recovering G from $(r^*,\phi ^*,\boldsymbol {\alpha }^*)$ . We now transfer the information we have gleaned about $(r,\phi ,\boldsymbol {\alpha })$ back to G itself. The partition of $[r]$ induces a partition $X_0,X_1,\ldots ,X_{r^*}$ on $V(G)$ . For every $\chi \in \mathop {\mathrm {RL}}^{-1}((r,\phi ,\mathcal {U}))$ , we almost always have $\chi (xy) \in \phi ^*(ij)$ when $ij \in \binom {[r^*]}{2}$ and $x \in X_i, y \in X_j$ . In Claims 4.2 and 4.3, we show, for almost all these $\chi $ , that $\chi ^{-1}(c)[X_i,X_j]$ is regular. Indeed, if $\chi ^{-1}(c)[X_i,X_j]$ is not regular, then Lemma 4.4 implies that there is $c^* \in [s]$ and large sets $X \subseteq X_i$ and $Y \subseteq X_j$ between which the density of colour $c^*$ edges is significantly less than it is between $X_i$ and $X_j$ . But Corollary 4.8 implies that very few s-edge colourings of $G[X_i,X_j]$ are such that some colour class ( $c \in \phi ^*(ij)$ ) has size much less than the average $|\phi ^*(ij)|^{-1}$ .

The final step is to look inside the classes. We discretise by applying the regularity lemma again within each class (which is large by Lemma 2.8). This allows us to apply Lemma 2.5 to prove (iii).

1.5 Organisation

Most of the paper concerns optimal triples for Problem $Q_t$ rather than graphs. In Section 2, we collect some tools concerning these optimal triples, and some consequences of the extension property. The main result in Section 3 is the stability of optimal solutions given the extension property, which is the key component of the proof of our graph stability result, Theorem 1.4. In Section 4, we transfer statements on optimal triples to graphs, and prove Theorem 1.4, after stating and proving some tools concerning the regularity lemma. Then we prove Corollary 1.5. Section 5 concerns applications of Theorem 1.4, in particular, it contains the derivations of Theorems 1.7 and 1.9. We finish with some concluding remarks in Section 6.