Proof. Let ${\varepsilon }> 0,m_0 \in \mathbb {N}$ be the output of Lemma 2.4 applied with $s^{-1},k_c$ playing the roles of $\eta ,k$ for every $c \in [s]$ . Let $n_0 := 2s^2 m_0/\mu $ and $\delta := {\varepsilon }^2$ . By decreasing ${\varepsilon }$ and increasing $m_0$ if necessary, we may assume that $\delta \ll \mu $ .

Certainly, $\mathcal {LG}_{\gamma ,\delta }(G;\phi ,\mathcal {Y}) \supseteq \mathcal {P}_{\gamma ,\delta }(G;\phi ,\mathcal {Y})$ by definition. Suppose that there exists $\chi \in \mathcal {LG}_{\gamma ,\delta }(G;\phi ,\mathcal {Y})$ such that the $(\gamma ,\delta )$ -perfect condition fails at some $x \in V(G)$ . Since $\chi $ is $(\gamma ,\delta )$ -locally good, there exists $i \in [r]$ such that for all $j \in [r]\setminus \{ i \}$ , we have that $|\chi ^{-1}(c)[x,Y_j]|=(|\phi (ij)|^{-1} \pm \delta )|Y_j|$ and $d_G(x,Y_i) - |\chi ^{-1}(1)[x,Y_i]| < \delta n$ . We will show that for all $j \in [r]\setminus \{ i \}$ and $y \in N_G(x,Y_j)$ , we have $\chi (xy) \in \phi (ij)$ , and for all $y \in N_G(x,Y_i)$ , we have $\chi (xy) = 1$ .

Suppose first that there exists $j \in [r]\setminus \{ i \}$ and $y \in N_G(x,Y_j)$ such that $\chi (xy) =: c \notin \phi (ij)$ . Let $k := k_{c}$ and $J := \chi ^{-1}(c)$ . Now, $\phi ^{-1}(c)$ is maximally $K_{k}$ -free. Since $ij \notin \phi ^{-1}(c)$ , we have that $\phi ^{-1}(c) \cup \{ ij \}$ contains a copy of $K_{k}$ . So there exist $i_3,\ldots ,i_k \in [r]\setminus \{ i,j \}$ such that $i_1,\ldots ,i_k$ span a copy of $K_k$ in $\phi ^{-1}(c)$ , where $i_1 := i$ and $i_2 := j$ .

Since $\chi $ is $(\gamma ,\delta )$ -locally good, for all $\ell =3,\ldots ,k$ , we have, by the pairs condition, taking the union over all parts $X \subseteq Y_{i_\ell }$ , that

$$ \begin{align*}|N_J(x,Y_{i_\ell}) \cap N_J(y,Y_{i_\ell})| \geq (|\phi(ii_\ell)|^{-1}|\phi(ji_\ell)|^{-1} - \delta)|Y_{i_\ell}| \geq \frac{|Y_{i_\ell}|}{2s^2} \geq \frac{\mu n}{2s^2} \geq m_0. \end{align*} $$

Let $U_\ell := N_J(x,Y_{i_\ell }) \cap N_J(y,Y_{i_\ell })$ . Proposition 2.5 implies that $J[U_\ell ,U_{\ell '}]$ is a $(\sqrt {\delta },\geq \! s^{-1})$ -regular pair for all distinct $\ell ,\ell ' \in \{ 3,\ldots , k \}$ . Lemma 2.4 implies that J contains a copy of $K_{k-2}$ . Together with $x,y$ , we obtain a copy of $K_k$ in $\chi ^{-1}(c)$ , a contradiction.

Suppose instead that there is some $y \in N_G(x,Y_i)$ such that $\chi (xy) =: c \neq 1$ . Let $k := k_c$ and $J := \chi ^{-1}(c)$ . Since $r \geq k_2-1 \geq k-1$ , Proposition 2.9 implies that i lies in a copy of $K_{k-1}$ in the graph $\phi ^{-1}(c)$ . Let $i_1 := i$ and let $i_2,\ldots ,i_{k-1}$ be the other vertices in this copy. As before, setting $U_\ell := N_J(x,Y_{i_\ell }) \cap N_J(y,Y_{i_\ell })$ for all $\ell \in [2,k-1]$ , we have that $J[U_2,\ldots ,U_{k-1}]$ contains a copy of $K_{k-2}$ . Together with $x,y$ , this gives a copy of $K_k$ in J, a contradiction.

Proof. For brevity, write $Q := Q(\boldsymbol {k})$ , $R := R(\boldsymbol {k})$ , $F(G) := F(G;\boldsymbol {k})$ , and abbreviate similarly elsewhere. Lemma 2.2(i) implies that there exists $\mu> 0$ such that for all , we have that $\alpha _i^*> \mu $ for all $i \in [r^*]$ . Applying Lemma 2.2(ii) gives the constant ${\varepsilon }_0$ . We may assume that $\delta \ll {\varepsilon }_0,\mu ,1/R$ since a smaller $\delta $ yields a stronger conclusion. Apply Lemma 2.4 with $(2s)^{-1},k_c$ playing the roles of $\eta ,k$ for all $c \in [s]$ to obtain $m_0 \in \mathbb {N}$ and ${\varepsilon }'> 0$ such that its conclusions hold. Choose constants $\delta _0,\ldots ,\delta _5 \in \mathbb {R}$ such that $0 < \delta _0 \ll \ldots \ll \delta _5 \ll \delta $ . We may assume that ${\varepsilon }' \geq 3\sqrt {\delta }_1$ . Apply Theorem 1.4 with $\delta _0$ playing the role of $\delta $ to obtain $n_0 \in \mathbb {N}$ and ${\varepsilon }> 0$ . Without loss of generality, we may assume that $0 < 1/n_0 \ll {\varepsilon } \ll \delta _0$ and

(2.2) $$ \begin{align} \sqrt{\delta_1}\mu n_0 \geq 2m_0. \end{align} $$

We may further assume that the conclusion of Lemma 2.11 holds with $s,\boldsymbol {k},\mu /2$ playing the roles of inputs $s,\boldsymbol {k},\mu $ , and with outputs $n_0 \in \mathbb {N}, \delta _5> 0$ . Thus, our constants form the hierarchy

(2.3) $$ \begin{align} 0 < 1/n_0 \ll {\varepsilon} \ll \delta_0 \ll \ldots \ll \delta_5 \ll \delta \ll {\varepsilon}_0,\mu,1/R. \end{align} $$

Let $G = K(X_1,\ldots ,X_m)$ be a complete m-partite graph on $n \geq n_0$ vertices such that $\log F(G) \geq (Q-{\varepsilon })\binom {n}{2}$ . Note that $m < R$ (or $F(G)=0$ ). Let also

(2.4) $$ \begin{align} Y_0 := \bigcup_{i \in [m] : |X_i| \leq \delta_1^4 n}X_i. \end{align} $$

Given , let $\mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ be the set of partitions $\mathcal {Y} := (Y_1,\ldots ,Y_{r^*})$ of $V(G)\setminus Y_0$ such that $Y_0,\ldots ,Y_{r^*}$ is a coarsening of $X_1,\ldots ,X_m$ , and

$$ \begin{align*}\sum_{i \in [r^*]}|\,|Y_i|-\alpha_i^*n\,| < \delta_1^2n. \end{align*} $$

(We do not yet know that $\mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ is nonempty.)

Given $\eta>0$ , let $\mathcal {G}_{\eta }(G)$ be the set of $\boldsymbol {k}$ -valid colourings $\chi $ of G for which there exist and $\mathcal {Y} \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ such that $\chi \in \mathcal {G}_{\eta }(G;\phi ^*,\mathcal {Y})$ . Call the elements of $\mathcal {G}_{\eta }(G) \eta $ -good. Given $\eta _1,\eta _2$ , let $\mathcal {LG}_{\eta _1,\eta _2}(G)$ and $\mathcal {P}_{\eta _1,\eta _2}(G)$ be as $\mathcal {G}_{\eta }(G)$ but with $\mathcal {LG}_{\eta _1,\eta _2}(G;\phi ^*,\mathcal {Y})$ and $\mathcal {P}_{\eta _1,\eta _2}(G;\phi ^*,\mathcal {Y})$ replacing $\mathcal {G}_{\eta }(G;\phi ^*,\mathcal {Y})$ , respectively.

Proof of Claim

By Theorem 1.4 applied to G with parameter $\delta _0$ , we have that there are at least $(1-2^{-{\varepsilon } n^2}) \cdot F(G)$ colourings $\chi : E(G) \rightarrow [s]$ which are $\boldsymbol {k}$ -valid and for which there is some and a partition $V_1 \cup \ldots \cup V_{r^*} = V(G)$ such that

• ○ For all $i \in [r^*]$ , we have $|\,|V_i|-\alpha ^*_i n\,| \leq 1$ .

• ○ $\chi $ is $\delta _0$ -good with respect to $(\phi ^*;V_1,\ldots ,V_{r^*})$ .

• ○ Suppose $\sum _{i \in [r^*]}e(G[V_i])> \delta _0 n^2$ . Then $\boldsymbol {k}$ does not have the strong extension property, and all but at most $\delta _0 n^2$ edges in $\bigcup _{i \in [r^*]}G[V_i]$ are coloured with $1$ under $\chi $ . Moreover, if $\boldsymbol {\ell } := (\ell _1,\ldots ,\ell _{r^*}) \in \mathbb {N}^{r^*}$ is such that at least $\delta _0|V_i|^2$ edges need to be removed from $G[V_i]$ to make it $K_{\ell _i}$ -free, then $\|\boldsymbol {\ell }\|_1 \leq k_{1}-1$ .

We will show that every such $\chi $ lies in $\mathcal {G}_{\delta _1}(G)$ , which implies the first part of the claim. Fix $\chi $ and its associated $(r^*,\phi ^*,\boldsymbol {\alpha }^*)$ and $V_1,\ldots , V_{r^*}$ (recall that both $(r^*,\phi ^*,\boldsymbol {\alpha }^*)$ and the partition $V_1\cup \ldots \cup V_{r^*}$ may be different for different $\chi $ ). For all $ij \in \binom {[r^*]}{2}$ , the $\delta _0$ -goodness of $\chi $ implies that

$$ \begin{align*}e_G(V_i,V_j) \geq \sum_{c \in \phi^*(ij)}|\chi^{-1}(c)[V_i,V_j]| \geq \sum_{c \in \phi^*(ij)}(|\phi^*(ij)|^{-1} - \delta_0)|V_i||V_j| \geq (1-s\delta_0)|V_i||V_j|, \end{align*} $$

so $e_{\overline {G}}(V_i,V_j) \leq s\delta _0n^2$ . Suppose that $X_k$ , some $k \in [m]$ , is such that $|X_k \cap V_i|> \sqrt {s\delta _0}n$ . Then, for every $j \in [m]\setminus \{k\}$ ,

$$ \begin{align*}|X_k \cap V_j| = \frac{e_{\overline{G}[X_k]}(V_i,V_j)}{|X_k \cap V_i|} \leq \frac{e_{\overline{G}}(V_i,V_j)}{|X_k \cap V_i|} < \sqrt{s\delta_0}n. \end{align*} $$

So for all $k \in [m]$ with $|X_k|> \delta _1^4n > R\sqrt {s\delta _0}n$ , there is a unique $i_k \in [r^*]$ such that $|X_k \cap V_{i_k}|> \sqrt {s\delta _0}n$ . For all $i \in [r^*]$ , let

$$ \begin{align*}Y_i := \bigcup_{k \in [m] : i_k = i}X_k. \end{align*} $$

Recall that we already defined $Y_0$ in (2.4). Then $Y_0,\ldots ,Y_{r^*}$ is a coarsening of $X_1,\ldots ,X_m$ , and $|X_j| \leq \delta _1^4 n$ if and only if $X_j \subseteq Y_0$ . For all $i \in [r^*]$ , we have that

(2.5) $$ \begin{align} |Y_i \bigtriangleup V_i| &\leq |Y_0| + \sum_{k \in [m]: i_k \neq i}|X_k \cap V_i| + \sum_{k \in [m]: i_k = i}|X_k \setminus V_i| \leq m\delta_1^4n + 2Rm\sqrt{s\delta_0}n \leq \delta_1^3 n \end{align} $$

and

(2.6) $$ \begin{align} \sum_{i \in [r^*]}|\,|Y_i| - \alpha^*_in \,| \leq \sum_{i \in [r^*]}\left(|\, |V_i| - |Y_i|\, | + |\,|V_i|-\alpha^*_i n\,| \right) \leq R\delta_1^3 n + R < \delta_1^{2} n, \end{align} $$

so $(Y_1,\ldots ,Y_{r^*}) \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha ^*})$ . Moreover, for all $i \in [r^*]$ , we have

(2.7) $$ \begin{align} |Y_i| \geq (\alpha^*_i - \delta^2_1)n-1 \geq \frac{\mu n}{2}. \end{align} $$

It remains to prove that $\chi $ is $\delta _1$ -good with respect to $(\phi ^*;Y_1,\ldots ,Y_{r^*})$ . Proposition 2.6 and (2.5) imply that for all $ij \in \binom {[r^*]}{2}$ and all $c \in \phi ^*(ij)$ , we have that $\chi ^{-1}(c)[Y_i,Y_j]$ is a $(\delta _1,|\phi ^*(ij)|^{-1})$ -regular pair. Moreover,

$$ \begin{align*} \phantom{=}& \sum_{i \in [r^*]}|\, e_G(Y_i) - |\chi^{-1}(1)[Y_i]| \,| \leq \sum_{i \in [r^*]}|\, e_G(V_i) - |\chi^{-1}(1)[V_i]| \,| + \sum_{i \in [r^*]}|Y_i \triangle V_i|^2\\ &\stackrel{({2.5})}{\leq} \delta_0 n^2 + R\delta_1^6 n^2 < \delta_1^5 n^2. \end{align*} $$

Thus, $\chi \in \mathcal {G}_{\delta _1}(G)$ . This completes the proof of the first part of the claim.

For the second part, fix , $\mathcal {Y} \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ and $\chi \in \mathcal {G}_{\delta _1}(G;\phi ^*,\mathcal {Y})$ . For each $i \in [r^*]$ , let $t_i$ be the number of parts $X_j$ which lie in $Y_i$ . Then $t_i$ is at most the number of parts $X_j$ which have intersection at least $\sqrt {s\delta _0}n$ with $V_i$ . Suppose first that $\sum _{i \in [r^*]}e_G(V_i) \leq \delta _0 n^2$ . If $t_i \geq 2$ for some $i \in [r^*]$ , then $e_G(V_i) \geq s\delta _0 n^2$ , a contradiction. So $t_1 = \ldots = t_{r^*} = 1$ .

Suppose now that $\sum _{i \in [r^*]}e_G(V_i)> \delta _0 n^2$ . Then by Theorem 1.4(iii), $\boldsymbol {k}$ does not have the strong extension property. Since $G[V_i]$ is a complete multipartite graph containing at least $t_i$ parts of size at least $\sqrt {s\delta _0}n$ , we have that at least $s\delta _0 n^2> \delta _0|V_i|^2$ edges need to be removed from $G[V_i]$ to make it $K_{t_i}$ -free. Thus, $t_1+\ldots +t_{r^*} \leq k_1-1$ , proving the second part of the claim.

This proves parts (i) and (ii) of Theorem 2.12. Namely, take any $\chi \in \mathcal {G}_{\delta _1}(G) \neq \emptyset $ . Let and $\mathcal {Y} := (Z_0,\ldots ,Z_{r^*}) \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ witness $\chi \in \mathcal {G}_{\delta _1}(G;\phi ^*,\mathcal {Y})$ . Then they satisfy items (i) and (ii) of Theorem 2.12.

Suppose now that (2.1) holds, but

(2.8) $$ \begin{align} |\mathcal{G}_{\delta_1}(G) \setminus \mathcal{LG}_{\delta_4,\delta_5}(G)|> 2^{-\delta_2 n} \cdot F(G). \end{align} $$

For most of the next part of the proof, we will establish a contradiction to this assumption. (Recall that, by Lemma 2.11, a direct contradiction to the statement of the theorem would replace $\mathcal {LG}_{\delta _4,\delta _5}(G)$ by $\mathcal {LG}_{0,\delta _5}(G)$ .)

For every , Lemma 2.1(iii) implies that $(\phi ^*)^{-1}(c)$ is maximally $K_{k_c}$ -free for all $c \in [s]$ , and that $r^* \geq k_2-1$ . Lemma 2.11 and (2.7) imply that for every $\mathcal {Y} \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ , we have that $\mathcal {P}_{\delta _4,\delta _5}(G;\phi ^*,\mathcal {Y}) = \mathcal {LG}_{\delta _4,\delta _5}(G;\phi ^*,\mathcal {Y})$ . So

(2.9) $$ \begin{align} \mathcal{LG}_{\delta_4,\delta_5}(G) = \mathcal{P}_{\delta_4,\delta_5}(G). \end{align} $$

Given $\mathcal {Y} = (Y_1,\ldots ,Y_{r^*}) \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ and $x \in V(G)$ , write $\mathcal {Y}-x$ to denote the partition $(Y_1\setminus \{x\},\ldots ,Y_{r^*}\setminus \{x\})$ and define $\mathcal {Y}-y-z$ similarly.

Proof of Claim

Suppose first that for at least half of the colourings in $\mathcal {G}_{\delta _1}(G)\setminus \mathcal {LG}_{\delta _4,\delta _5}(G)$ , the locally good condition fails at some vertex (rather than only at pairs). We will show that there exists an x satisfying (i) and (ii). By (2.8) and (2.9), there is some $x \in V(G)$ such that there are at least $\frac {1}{2}\cdot \frac {1}{n} \cdot 2^{-\delta _2 n} \cdot F(G) \geq 2^{-2\delta _2 n} \cdot F(G)$ valid colourings $\chi $ which are $\delta _1$ -good (with respect to some optimal solution and partition) but for which the $(\delta _4,\delta _5)$ -locally good condition fails at x (for all optimal solutions and partitions). Call this set of colourings $\mathcal {L}_x(G)$ .

We have $\log F(G-x) \geq \log (s^{-n}\cdot F(G)) \geq (Q-{\varepsilon })\binom {n}{2} - n\log s \geq (Q-2{\varepsilon })\binom {n-1}{2}$ , so a version of Claim 2.12.1(i) applied to $G-x$ implies that

$$ \begin{align*}|\mathcal{G}_{2\delta_1}(G-x)| \geq (1-2^{-2{\varepsilon} (n-1)^2})\cdot F(G-x). \end{align*} $$

Suppose that x does not satisfy Claim 2.12.2. Then for each $\boldsymbol {k}$ -valid colouring $\chi $ of $G-x$ , either it does not lie in $\mathcal {G}_{2\delta _1}(G-x)$ or it does lie in $\mathcal {G}_{2\delta _1}(G-x)$ but has at most $2^{(Q-\delta _3)n} \boldsymbol {k}$ -valid extensions which lie in $\mathcal {L}_x(G)$ . We have that

$$ \begin{align*} 2^{-2\delta_2n}\cdot F(G) &\leq |\mathcal{L}_x(G)| \leq |\mathcal{G}_{2\delta_1}(G-x)| \cdot 2^{(Q-\delta_3)n} + (F(G-x)-|\mathcal{G}_{2\delta_1}(G-x)|) \cdot s^n\\ &\leq F(G-x)\cdot 2^{(Q-\delta_3)n} + s^{n} \cdot 2^{-2{\varepsilon} (n-1)^2}F(G-x)\\ &\leq F(G-x)\cdot 2^{(Q-\delta_3/2)n}, \end{align*} $$

and so

$$ \begin{align*}\log F(G) - \log F(G-x) \leq (Q - \delta_3/2 + 2\delta_2)n \leq (Q-\delta_3/3)n < (Q-{\varepsilon})n, \end{align*} $$

a contradiction to (2.1).

Thus, we may assume that at least half the good but not locally good colourings fail due to the pair condition. Again, there is a pair $y,z$ of distinct vertices that appear in at least $2^{-2\delta _2 n}\cdot F(G)$ such colourings, and an identical argument gives the required $\xi $ , satisfying (iii) and (iv).

Suppose there exist $x,\chi $ as in Claim 2.12.2(i) and (ii). So there are and $\mathcal {Y} = (Y_1,\ldots ,Y_{r^*}) \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ and $\mathrm {Ext}(\chi )$ as in Claim 2.12.2. Define $\boldsymbol {\beta } := (\beta _1,\ldots ,\beta _{r^*})$ by setting $\beta _i := |Y_i|/n$ for all $i \in [r^*]$ . Then (2.6) implies that

(2.10) $$ \begin{align} \|\boldsymbol{\beta} - \boldsymbol{\alpha}^*\|_1 < \delta_1^2. \end{align} $$

Note that $\mathrm {Ext}(\chi ) \subseteq \mathcal {G}_{3\delta _1}(G;\phi ^*,\mathcal {Y})$ . For every $\overline {\chi } \in \mathrm {Ext}(\chi )$ , define $\phi = \phi (\overline {\chi }) : \binom {[r^*+1]}{2} \rightarrow 2^{[s]}$ by setting

$$ \begin{align*}\phi(ij) := \begin{cases} \phi^*(ij) &\mbox{if } ij \in \binom{[r^*]}{2}; \\ \{ c \in [s] : |\overline{\chi}^{-1}(c)[x,Y_j]| \geq \sqrt{\delta_1} |Y_{j}| \} &\mbox{if } i = r^*+1, j \in [r^*]. \\ \end{cases} \end{align*} $$

Fix the pattern $\phi $ that appears for the largest number of extensions $\overline {\chi } \in \mathrm {Ext}(\chi )$ , and let $\mathrm {Ext}_{\phi }(\chi )$ be the set of these $\overline {\chi }$ . By Claim 2.12.2(ii),

(2.11) $$ \begin{align} |\mathrm{Ext}_{\phi}(\chi)| \geq 2^{-sr^*} \cdot |\mathrm{Ext}(\chi)| \geq 2^{(Q-2\delta_3)n}. \end{align} $$

Proof of Claim

Suppose not. Then there is some $c \in [s]$ such that $\phi ^{-1}(c)$ contains a copy of $K_{k_c}$ . Since $\phi ^* \in \Phi _2(r^*;\boldsymbol {k})$ is the restriction of $\phi $ to $\binom {[r^*]}{2}$ , the vertex $r^*+1$ must lie in this copy. Let $z_1,\ldots ,z_{k_c-1}$ be the other vertices. Let $\overline {\chi } \in \mathrm {Ext}_{\phi }(\chi )$ be arbitrary. For each $k \in [k_c-1]$ , let $Z_k := \{ y \in Y_{z_k} : \overline {\chi }(xy) = c \}$ . Then, by the definition of $\phi $ , we have

(2.12) $$ \begin{align} |Z_k|> \sqrt{\delta_1}|Y_{z_k}| \stackrel{({2.2})}{\geq} m_0. \end{align} $$

Let $kk' \in \binom {[k_c-1]}{2}$ . Now, $\overline {\chi }^{-1}(c)[Y_{z_k},Y_{z_{k'}}]$ is $(3\delta _1,\geq \!\! s^{-1})$ -regular since $\overline {\chi } \in \mathrm {Ext}(\chi ) \subseteq \mathcal {G}_{3\delta _1}(G; \phi ^*,\mathcal {Y})$ . Therefore,

$$ \begin{align*}d(\overline{\chi}^{-1}(c)[Z_k,Z_{k'}]) \geq d(\chi^{-1}(c)[Y_{z_k},Y_{z_{k'}}]) - 3\delta_1 \geq |\phi^*(z_kz_{k'})|^{-1} - 6\delta_1 \geq 1/(2s). \end{align*} $$

Now Proposition 2.7 implies that $\overline {\chi }^{-1}(c)[Z_k,Z_{k'}]$ is $3\sqrt {\delta _1}$ -regular. Therefore, $\overline {\chi }^{-1}(c)[Z_k,Z_{k'}]$ is $(3\sqrt {\delta _1},\geq \!\!1/(2s))$ -regular. Now (2.12) and Lemma 2.4 imply that $\chi ^{-1}(c)[Z_1,\ldots ,Z_{k_c-1}]$ contains a copy of $K_{k_c-1}$ . Together with x, this gives a c-coloured copy of $K_{k_c}$ in $\overline {\chi }$ , contradicting the fact that $\overline {\chi }$ is $\boldsymbol {k}$ -valid.

Next we will show that there is some $i \in [r^*]$ such that $r^*+1$ is a clone of i under $\phi $ . Suppose for a contradiction that this is not the case. Thus, by our choice of ${\varepsilon }_0$ , we have $\mathop {\mathrm {ext}}(\phi ,\boldsymbol {\alpha }^*) \leq Q-{\varepsilon }_0$ . Then, using (a version of) Proposition 2.1(i) and Lemma 2.2(ii),

(2.13) $$ \begin{align} \frac{1}{n}\log\prod_{j \in [r^*]}|\phi(\{j,r^*+1\})|^{|Y_j|} &= \mathop{\mathrm{ext}}(\phi,\boldsymbol{\beta}) \leq \mathop{\mathrm{ext}}(\phi,\boldsymbol{\alpha}^*) + \|\boldsymbol{\alpha}^*-\boldsymbol{\beta}\|_1 \cdot \log s\\ \nonumber &\stackrel{({2.10})}{<} Q - {\varepsilon}_0 + \delta_1^2\log s \leq Q - {\varepsilon}_0/2. \end{align} $$

The number of possible patterns $\phi $ on $r^*+1$ vertices which are extensions of $\phi ^*$ is at most $2^{sr^*}$ , and, by definition, the number of edges with endpoint x which are not coloured according to $\phi $ is at most $s\sqrt {\delta _1}n$ . Therefore, the total number of $\boldsymbol {k}$ -valid extensions $\overline {\chi } \in \mathrm {Ext}(\chi )$ of $\chi $ to G is at most

(2.14) $$ \begin{align} 2^{sr^*} \cdot \binom{n}{\leq s\sqrt{\delta_1}n} \cdot s^{s\sqrt{\delta_1}n} \cdot s^{|Y_0|} \cdot \prod_{j \in [r^*]} |\phi(\{ j,r^*+1 \})|^{|Y_j|} \stackrel{({2.4}),({2.13})}{\leq} 2^{(Q-{\varepsilon}_0/3)n} < 2^{(Q-2\delta_3)n}. \end{align} $$

So certainly there are at most this number of extensions which lie in $\mathrm {Ext}_{\phi }(\chi )$ , contradicting (2.11).

Therefore, we may assume that there is some $i^* \in [r^*]$ such that $r^*+1$ is a clone of $i^*$ under $\phi $ . Thus, we have

(2.15) $$ \begin{align} \phi(\{ j,r^*+1 \}) = \phi^*(i^*j) \ \text{ for all } \ j \in [r^*]\setminus \{ i^*\} \ \text{ and } \ \phi(\{ i^*,r^*+1 \}) \subseteq \{ 1 \}, \end{align} $$

where the last inclusion follows from Lemma 2.2(iii).

Proof of Claim

Define $\ell \in \{ 0,\ldots , r^* \}$ as follows. If there is $i' \in [r^*]$ such that $d_G(x,Y_{i'}) < (1-\delta _4)|Y_{i'}|$ (noting that there can be at most one such $i'$ ), let $\ell := i'$ . If there is no such $i'$ , let $\ell :=0$ . Let X be the part of G which contains x.

Case 1: $\ell = 0$ .

Suppose that (i) is false. Since $\overline {\chi }$ is not $(\delta _4,\delta _5)$ -locally good at x, we must have that $d_G(x,Y_{i^*}) - |\overline {\chi }^{-1}(1)[x,Y_{i^*}]|> \delta _5 |Y_{i^*}|$ . By (2.15) and the definition of $\phi $ , we have that

(2.16) $$ \begin{align} d_G(x,Y_{i^*}) - |\overline{\chi}^{-1}(1)[x,Y_{i^*}]| &= \sum_{c \in \{ 2,\ldots, s \}}|\overline{\chi}^{-1}(c)[x,Y_{i^*}]| < (s-1)\sqrt{\delta_1}|Y_{i^*}| < \delta_5 |Y_{i^*}|, \end{align} $$

a contradiction. For (ii), let $X \subseteq Y_h$ be the part of G containing x and suppose $h \in [r^*]$ . Then $|Y_h|-|X|=d_G(x,Y_h) \geq (1-\delta _4)|Y_h|$ . So in this case, $|X| \leq \delta _4|Y_h|$ and (ii) is vacuous.

Case 2: $\ell \in [r^*]$ .

We will first show that (ii) holds. Note that $h=\ell $ since $X \subseteq Y_h$ and $x\in X$ is not adjacent to the whole of $Y_\ell $ . If $Y_{h} = X$ , then $N_G(x,Y_{h}) = \emptyset $ so $\phi (\{ r^*+1,h \}) = \emptyset $ . So (2.15) implies that $h = i^*$ , as required.

Otherwise, $Y_{h} \neq X$ . Suppose that $i^* \neq h$ . Then $|\phi ^*(i^*h)| \geq 2$ , and so

$$ \begin{align*} \frac{1}{n}\left( \log \prod_{j \in [r^*]} |\phi(\{ j,r^*+1 \})|^{d_G(x,Y_j)}\right) &\stackrel{({2.15})}{=} \frac{1}{n}\sum_{j \in [r^*]\setminus \{ i^* \}} |Y_j\setminus X| \log|\phi^*(i^*j)|\\ &= \sum_{j \in [r^*]\setminus \{ i^*,h \}}\beta_j \log|\phi^*(i^*j)| + \left(\beta_h -\frac{|X|}{n}\right) \log|\phi^*(i^*h)|\\ &\leq q_{i^*}(\phi^*,\boldsymbol{\alpha}^*) + \left( \|\boldsymbol{\alpha}^*-\boldsymbol{\beta}\|_1 - \frac{|X|}{n}\right)\log s\\ &\stackrel{({2.10})}{\leq} Q + (\delta_1^2 - \delta_4^2)\log s < Q - 4\delta_3, \end{align*} $$

where the penultimate inequality follows from Proposition 2.1(ii). Combining this with a very similar calculation to (2.14) implies that the total number of $\boldsymbol {k}$ -valid extensions $\overline {\chi } \in \mathrm {Ext}(\chi )$ of $\chi $ to G is less than $2^{(Q-2\delta _3)n}$ , contradicting (2.11). Therefore, $i^*=h$ , proving (ii). The proof of (i) now proceeds exactly as in Case 1.

Claim 2.12.4(i) implies that for every $\overline {\chi } \in \mathrm {Ext}_{\phi }(\chi )$ , there exist $j^* \in [r^*]\setminus \{i^*\}$ , a part $X \subseteq Y_{j^*}$ and $c^* \in \phi ^*(i^*j^*)$ such that, by averaging,

(2.17) $$ \begin{align} |\overline{\chi}^{-1}(c^*)[x,X]| \leq |\phi^*(i^*j^*)|^{-1}|X|-\frac{\delta_5}{s\ell_{j^*}}|Y_{j^*}|. \end{align} $$

In particular, since $\ell _{j^*} \leq k_1 < R$ , we have $|X| \geq \delta _5|Y_{j^*}|/(s \cdot R) \geq (\delta _5)^2|Y_{j^*}|$ . The number of ways of adding edges coloured with $\phi ^*$ between x and X with this reduced density (choosing the set of colour- $c^*$ neighbours, and then colouring every other edge with any available colour other than $c^*$ ) is at most

$$ \begin{align*} &\phantom{=} \sum_{k=0}^{\lfloor |\phi^*(i^*j^*)|^{-1}|X|-\frac{\delta_5}{s\ell_{j^*}}|Y_{j^*}|\rfloor} \binom{|X|}{k}(|\phi^*(i^*j^*)|-1)^{|X|-k}\\ &\leq e^{-\delta_5^2|X|/3(s^2R^2)} \cdot |\phi^*(i^*j^*)|^{|X|} \stackrel{({2.7})}{\leq} e^{-\delta_5^5 n} \cdot |\phi^*(i^*j^*)|^{|X|}, \end{align*} $$

where the first inequality is a consequence of Lemma 2.8. Using Proposition 2.1(ii), we have that

(2.18) $$ \begin{align} q_{i^*}(\phi^*,\boldsymbol{\beta}) \leq Q + \|\boldsymbol{\alpha}^*-\boldsymbol{\beta}\|_1 (2\log s) \stackrel{({2.10})}{\leq} Q+2\delta_1^2\log s. \end{align} $$

Now, we can generate every $\overline {\chi } \in \mathrm {Ext}_{\phi }(\chi )$ from $\chi $ by doing the following. For each $j \in [r^*]$ , choose at most $\sqrt {\delta _1}|Y_j|$ vertices $y \in N_G(x,Y_j)$ and colour $xy$ arbitrarily. Arbitrarily colour all edges $xy_0$ where $y_0 \in Y_0$ . Then choose $j^*,X,c^*$ as above and colour every uncoloured edge (with endpoint x) according to $\phi $ so that (2.17) holds (with a small adjustment to account for the $\sqrt {\delta _1}|Y_j|$ edges that have already been coloured). That this will indeed generate every $\overline {\chi } \in \mathrm {Ext}_{\phi }(\chi )$ is a consequence of the definition of $\phi $ and Claim 2.12.4. Therefore,

$$ \begin{align*} |\mathrm{Ext}_{\phi}(\chi)| &\leq \binom{n}{\leq \sqrt{\delta_1}n} \cdot s^{\sqrt{\delta_1}n} \cdot s^{|Y_0|} \cdot e^{-\delta_5^5 n} \cdot \prod_{j \in [r^*]\setminus \{ i^*\}} |\phi^*(i^*j)|^{|Y_j|}\\ &\stackrel{({2.4}),({2.18})}{\leq} 2^{-\delta_5^6 n} \cdot 2^{(Q+2\delta_1^2 \log s)n} < 2^{(Q-2\delta_3)n}, \end{align*} $$

contradicting (2.11). So our assumption that there are $x,\chi $ as in Claim 2.12.2(i) and (ii) is false.

Thus, there must be $z,y,\xi $ such that Claim 2.12.2(iii) and (iv) hold. So there are and $\mathcal {Y} = (Y_1,\ldots ,Y_{r^*}) \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ and $\mathrm {Ext}(\xi )$ as in Claim 2.12.2. Again, we define colour patterns $\phi _y, \phi _z$ in analogy with $\phi $ to be the pair of patterns that appear together for the greatest number of extensions $\overline {\xi } \in \mathrm {Ext}(\xi )$ , and we write $\mathrm { Ext}_{\phi _y,\phi _z}(\xi )$ for the set of extensions $\overline {\xi } \in \mathrm {Ext}(\xi )$ with this pair of patterns. Again, by Claim 2.12.2(iii),

(2.19) $$ \begin{align} |\mathrm{Ext}_{\phi_y,\phi_z}(\xi)| \geq 2^{(Q-2\delta_3)(n+n-1)}. \end{align} $$

Similarly to Claim 2.12.3, we have $\phi _y,\phi _z \in \Phi _0(r^*+1;\boldsymbol {k})$ . Applying Lemma 2.2(ii), we have that there are $i_y, i_z \in [r^*]$ such that in $\phi _y,\phi _z$ , respectively, the vertex $r^*+1$ is a clone of $i_y,i_z$ . And, in particular, $\phi _y(\{i_y,r^*+1\}) \subseteq \{1\}$ and $\phi _z(\{i_z,r^*+1\}) \subseteq \{1\}$ . The analogue of Claim 2.12.4(i) for pairs holds in the same way; that is, there exist $j^* \in [r^*]\setminus \{i_y,i_z\}$ , a part $X \subseteq Y_{j^*}$ , $c_y \in \phi _y(i_yj^*)$ and $c_z \in \phi _z(i_zj^*)$ such that

$$ \begin{align*}|N_{\xi^{-1}(c_y)}(y,X) \cap N_{\xi^{-1}(c_z)}(z,X)| \neq |\phi^*(i_yj^*)|^{-1}|\phi^*(i_zj^*)|^{-1}|X| \pm \frac{\delta_5}{\ell_{j^*}}|Y_{j^*}|. \end{align*} $$

It remains to show that this implies there are few extensions of $\chi $ to G. By averaging, there are $c_y^* \in \phi ^*(i_yj^*)$ and $c_z^* \in \phi ^*(i_zj^*)$ such that

$$ \begin{align*}|N_{\xi^{-1}(c_y^*)}(y,X) \cap N_{\xi^{-1}(c_z^*)}(z,X)| < |\phi^*(i_yj^*)|^{-1}|\phi^*(i_zj^*)|^{-1}|X| - \frac{\delta_5}{s^2\ell_{j^*}}|Y_{j^*}|. \end{align*} $$

The number of ways of adding edges coloured with $\phi ^*$ between $y,z$ and X with this reduced density (choosing the set of v such that $yv$ is coloured $c_y^*$ and $zv$ is coloured $c_z^*$ , and then colouring every other $yu,zu$ with any pair other than $c_y^*,c_z^*$ ) is at most

$$ \begin{align*} &\phantom{=} \sum_{k=0}^{\lfloor |\phi^*(i_yj^*)|^{-1}|\phi^*(i_zj^*)|^{-1}|X|-\frac{\delta_5}{s^2\ell_{j^*}}|Y_{j^*}|\rfloor} \binom{|X|}{k}(|\phi^*(i_yj^*)||\phi^*(i_zj^*)|-1)^{|X|-k} \cdot s\\ &\hspace{4cm}\stackrel{({2.7})}{\leq} e^{-\delta_5^5 n} \cdot (|\phi^*(i_yj^*)||\phi^*(i_zj^*)|)^{|X|}. \end{align*} $$

Repeating the calculations in the single vertex case, we see that $|\mathrm {Ext}_{\phi _y,\phi _z}(\xi )| < 2^{(Q-2\delta _3)(n+n-1)}$ , contradicting 2.19).

Thus, our assumption (2.8) is false. Therefore, combining its negation with Claim 2.12.1, we see that

$$ \begin{align*} |\mathcal{P}_{\delta_4,\delta_5}(G)| \geq |\mathcal{G}_{\delta_1}(G)| - 2^{-\delta_2 n} \cdot F(G) \geq (1-2^{-{\varepsilon} n}) \cdot F(G). \end{align*} $$

Let $\chi \in \mathcal {P}_{\delta _4,\delta _5}(G)$ . Then there exists and $\mathcal {Y} = (Y_1,\ldots ,Y_{r^*}) \in \mathop {\mathrm {Coars}}(\boldsymbol {\alpha }^*)$ such that $\chi $ is $(\delta _4,\delta _5)$ -perfect with respect to $(\phi ^*;Y_1,\ldots ,Y_{r^*})$ . So for all $x \in V(G) \setminus Y_0$ , there exists $i(x) \in [r^*]$ such that for all $j \in [r^*]$ and $b \in N_G(x,Y_j)$ , we have $\chi (xb) = 1$ if $j = i(x)$ , and $\chi (xb) \in \phi ^*(i(x)j)$ otherwise. Moreover, (the proof of) the second part of Claim 2.12.4 implies that, if there exists $h(x) \in [r^*]$ such that the part X of G containing x lies in $Y_{h(x)}$ , and $|X|> \delta _4|Y_{h(x)}|$ , then $i(x) = h(x)$ .

Now, define a new partition $Z_0,\ldots ,Z_{r^*}$ of $V(G)$ by setting, for all $i \in [r^*]$ ,

$$ \begin{align*}Z_i := \{ x \in V(G) : i(x) = i \text{ and }x \in X \subseteq Y_{h(x)} \text{ with }|X| \geq \delta_4|Y_{h(x)}| \} \quad\text{and}\quad Z_0 := V(G) \setminus \textstyle\bigcup_{i \in [r^*]}Z_i. \end{align*} $$

Then $Z_0,\ldots ,Z_{r^*}$ is a coarsening of $X_1,\ldots ,X_m$ , and $\chi $ is $(0,\delta _5)$ -perfect with respect to $(\phi ^*;Z_1,\ldots ,Z_{r^*})$ by definition. Moreover, for all $i \in [r^*]$ , $Z_i \subseteq Y_i$ , and $|Y_i \setminus Z_i| \leq R\delta _4|Y_i|$ . So

$$ \begin{align*}\sum_{i \in [r^*]}|\, |Z_i| - \alpha^*_in\, | \leq \sum_{i \in [r^*]}(|Y_i| - |Z_i|) + \sum_{i \in [r^*]}|\, |Y_i| - \alpha^*_in \,| \leq R\delta_4 n + \delta_1^2 n < \delta n, \end{align*} $$

as required. Finally, let $\ell _i$ be the number of $X_j$ in $Z_i$ . Then $\ell _i \leq t_i$ for all $i \in [r^*]$ . The second part of Claim 2.12.1 yields the desired conclusion.

Proof. Since G is not complete multipartite, there exist distinct nonadjacent vertices $u,v \in V(G)$ such that $N_G(u) \neq N_G(v)$ . For any graph H, let $\chi (H)$ be the set of $\boldsymbol {k}$ -valid colourings of H. For each $\chi \in \chi (G-u-v)$ , let $\chi _u, \chi _v$ denote the number of valid extensions of $\chi $ to $G-v$ and $G-u$ , respectively. Since u and v are nonadjacent,

$$ \begin{align*}F(G;\boldsymbol{k}) = \sum_{\chi \in \chi(G-u-v)}\chi_u\chi_v. \end{align*} $$

Let $G_u$ denote the graph obtained from G by replacing v by a twin of u. Define $G_v$ similarly. The operation of passing from G to $G_u$ or $G_v$ is a symmetrisation. We have

$$ \begin{align*}F(G_u;\boldsymbol{k}) = \sum_{\chi \in \chi(G-u-v)}\chi_u^2 \ \text{ and } \ F(G_v;\boldsymbol{k}) = \sum_{\chi \in \chi(G-u-v)}\chi_v^2. \end{align*} $$

Then

$$ \begin{align*}0 \geq F(G_u;\boldsymbol{k}) + F(G_v;\boldsymbol{k}) - 2F(G;\boldsymbol{k}) = \sum_{\chi \in \chi(G-u-v)} (\chi_u - \chi_v)^2 \geq 0. \end{align*} $$

Therefore, $G_u$ and $G_v$ are both $\boldsymbol {k}$ -extremal.

Let $\mathcal {G}$ be the directed graph whose vertex set contains all n-vertex graphs (up to isomorphism) that can be obtained from G by a sequence of symmetrisations and add a directed edge from H to $H' \neq H$ if $H'\cong H_u$ for some vertex u. Note that H has outdegree equal to $0$ in $\mathcal {G}$ if and only if H is complete multipartite. By [Reference Pikhurko, Staden and Yilma11, Theorem 3], there is at least one sequence of symmetrisations which leads to a ( $\boldsymbol {k}$ -extremal) complete partite graph H; among all choices, pick one such that the number m of parts in H is as small as possible. We may assume that every part has size at least two, or we are done. Let $H^-$ be an inneighbour of H in $\mathcal {G}$ . Observe that $H^-$ is not complete multipartite since it does not have $0$ outdegree. Then there exists $x \in V(H^-)$ such that $H^--x$ is a complete m-partite graph with parts $V_1,\ldots ,V_m$ .

Therefore, x has a neighbour in each of $V_1,\ldots ,V_m$ . Then, for each $i \in [m]$ , $V_i$ has partition $A_i,B_i$ , where $xa \in E(H^-)$ for all $a \in A_i$ , and $xb \notin E(H^-)$ for all $b \in B_i$ ; and $A_i \neq \emptyset $ . Observe that every $A_i$ is a set of twins, and $B_i$ is a set of twins. For each $i \in [m]$ , replace $B_i$ by a set of $|B_i|$ twins of $a \in A_i$ . Thus, obtain a graph $H' \cong K(\{ x \}, V_1,\ldots ,V_m)$ which is a vertex of $\mathcal {G}$ , as required.

Proof. As before, we omit $\boldsymbol {k}$ from our notation where it is clear from the context, so, for example, $F(H) := F(H;\boldsymbol {k})$ . Choose constants $\delta ,\eta $ such that $0 < \eta \ll \delta \ll \gamma ,{\varepsilon }$ . By (1.3), we can choose $n_0$ to be such that whenever $N \geq n_0/2$ , we have that

(2.22) $$ \begin{align} (Q - \eta)\binom{N}{2} \leq \log F(N) \leq (Q + \eta)\binom{N}{2}. \end{align} $$

Without loss of generality, we may suppose that $1/n_0 \ll \eta $ . Let G be a graph on $n \geq n_0$ vertices. Suppose that there is some $x \in V(G)$ which does not satisfy (2.21). Let $T \subseteq V(G)$ be the set of twins of x (including x). Let $\chi (G-T)$ be the set of $\boldsymbol {k}$ -valid colourings of $G-T$ and let $c(\chi ,x)$ be the number of extensions of $\chi $ to $G - (T \setminus \{ x \})$ . For each $\chi \in \chi (G-T)$ , since every pair of vertices of X are twins, we have that $c(\chi ,x) \equiv c_\chi $ for all $x \in T$ . Therefore, if we list the vertices of T as $x_1,\ldots ,x_{|T|}$ , and let $t \in [\,|T|\,]$ , we have that

(2.23) $$ \begin{align} F(G-x_1-\ldots-x_t) = \sum_{\chi \in \chi(G-T)}c_\chi^{|T|-t}. \end{align} $$

Choose t with $1 \leq t \leq |T|$ to be maximal such that

(2.24) $$ \begin{align} \log F(G-x_1-\ldots-x_{i-1}) < (Q-{\varepsilon})n + \log F(G-x_1 - \ldots - x_i) \end{align} $$

for all $i \in [t]$ . (Since (2.21) does not hold, $t:=1$ satisfies the inequality). Suppose $t < |T|$ . By the Cauchy-Schwarz inequality,

$$ \begin{align*} F(G-x_1-\ldots - x_t)^2 &= \left(\sum_{\chi \in \chi(G-T)}c_\chi^{|T|-t} \right)^2 \leq \left(\sum_{\chi \in \chi(G-T)} c_\chi^{|T|-t-1} \right) \left( \sum_{\chi \in \chi(G-T)} c_\chi^{|T|-t+1} \right)\\ &\stackrel{({2.26})}{=} F(G-x_1-\ldots-x_{t+1}) F(G-x_1-\ldots-x_{t-1})\\ &\stackrel{({2.24})}{<} 2^{(Q-{\varepsilon})n} F(G-x_1-\ldots-x_{t+1}) F(G-x_1-\ldots-x_t). \end{align*} $$

So $F(G-x_1-\ldots -x_t) < 2^{(Q-{\varepsilon })n} \cdot F(G-x_1-\ldots -x_{t+1})$ . Thus, (2.24) holds for $t+1$ which is a contradiction to the maximality of t. So $t=|T|$ . Therefore, inductively, for all nonempty $T' \subseteq T$ ,

$$ \begin{align*}\log F(G) \leq (Q-{\varepsilon})|T'|n + \log F(G-T'). \end{align*} $$

Choose $T' \subseteq T$ with $|T'| = \lceil \delta n\rceil $ . Then

$$ \begin{align*} \log F(G) &\leq (Q-{\varepsilon})|T'|n + \log F(G-T') \stackrel{({2.22})}{\leq} Q|T'|n -{\varepsilon} \delta n^2 + (Q+\eta)\binom{n-|T'|}{2}\\ &\leq Q\binom{n}{2} + (-{\varepsilon} \delta + \eta +Q\delta^2)n^2 < Q\binom{n}{2} + (\eta + \delta^2 \log s - {\varepsilon} \delta )n^2\\ &< (Q-\eta)\binom{n}{2}, \end{align*} $$

a contradiction to (2.22).

Proof of Theorem 1.6

Again, we omit $\boldsymbol {k}$ from our notation where possible. Let ${\varepsilon }> 0$ . Let $\mu> 0$ be the constant obtained from Lemma 2.2(i) and let ${\varepsilon }_0$ be the constant obtained from Lemma 2.2(ii). Without loss of generality, we may assume that $0 < {\varepsilon }_0 \ll \mu ,{\varepsilon },1/R,1/s$ . Choose $\delta _4,\delta _3,\delta _2,\delta _1>0$ in this order such that $\delta _3$ is at most the output of Corollary 2.13 with input $\delta _4$ , and similarly, $\delta _2$ from $\delta _3$ , and $\delta _1$ from $\delta _2^2$ . Further, let $n_0$ be at least the integer output of all of these applications. By increasing $n_0$ and decreasing $\delta _1,\ldots ,\delta _4$ , we may assume that $0<1/n_0 \ll \delta _1 \ll \ldots \ll \delta _4 \ll {\varepsilon }_0$ and for all $n \geq n_0$ , by (1.3),

(2.25) $$ \begin{align} \left(Q - \delta_1 \right)\binom{n}{2} \leq \log F(n) \leq \left(Q + \delta_1 \right)\binom{n}{2} \end{align} $$

and that Lemma 2.15 applied with $\delta _3,\mu /2$ playing the roles of ${\varepsilon },\gamma $ , respectively, has output $n_1 \leq n_0$ and $\eta> 2\delta _1$ . Altogether, we have the hierarchy