2. Definitions and key lemmas

In this section, we recall several standard definitions and note two key lemmas that will be needed in the proof of Theorem 1.1. Most of these revolve around the concept of sparse regularity (for a thorough overview of which we refer the reader to the survey by Gerke and Steger [Reference Gerke and Steger12]).

For ${\varepsilon } \gt 0$ and $p \in (0, 1]$ , a pair of sets $(V_1, V_2)$ is said to be $({\varepsilon }, p)$ -lower-regular in a graph $G$ if, for all $U_i \subseteq V_i$ , $i \in \{1,2\}$ , with $|U_i| \geqslant{\varepsilon }|V_i|$ , the density $d_G(U_1, U_2) = e_G(U_1, U_2)/(|U_1||U_2|)$ of edges between $U_1$ and $U_2$ satisfies

\begin{equation*} d_G(U_1, U_2) \geqslant (1-{\varepsilon })p. \end{equation*}

Immediately from this definition, we get that in every $({\varepsilon },p)$ -lower-regular pair $(V_1, V_2)$ , for each $i \in \{1,2\}$ , all but at most ${\varepsilon }|V_i|$ vertices in $V_i$ have degree at least $(1-{\varepsilon })p|V_{3-i}|$ into $V_{3-i}$ – a fact we will make use of in the proof of Theorem 1.1. Another useful and well-known property is that lower-regularity is inherited on large sets.

For $\lambda \gt 0$ and $p \in (0, 1]$ , a graph $G$ is said to be $(\lambda,p)$ -uniform if, for all disjoint $X, Y \subseteq V(G)$ with $|X|, |Y| \geqslant \lambda |V(G)|$ , the density of edges between $X$ and $Y$ satisfies $d_G(X,Y) = (1\pm \lambda )p$ . If only the upper bound holds, the graph is said to be upper-uniform.Footnote ¹ For example, it is easy to see that the random graph $G_{n, p}$ is with high probability $(o(1),p)$ -uniform whenever $p \gg 1/n$ . If $G = (V_1, V_2;\; E)$ is bipartite, we say that $G$ is $(\lambda, p)$ -uniform or upper-uniform if the same conditions hold for all $X \subseteq V_1$ and $Y \subseteq V_2$ with $|X| \geqslant \lambda |V_1|$ and $|Y| \geqslant \lambda |V_2|$ . In order to prove our main technical lemma, we rely on the following result, a simple corollary of [Reference Kohayakawa, Konstadinidis and Mota16, Lemma 6], whose proof follows a density increment argument. The same conclusion can also be obtained by an application of the sparse regularity lemma.

The next lemma is the crux of our argument. Here and elsewhere, we say that $(X,Y;\; E)$ is lower-regular if $(X,Y)$ is lower-regular with respect to the set of edges $E$ .

Proof. Given $\varepsilon$ , $r$ , and $\Delta$ , we let $\alpha = 1/(2r)$ , ${\varepsilon }_{\Delta +1} \;:\!=\;{\varepsilon }$ , $\lambda _{\Delta +1} = \lambda _{2.2}({\varepsilon }_{\Delta +1},\alpha )$ , and, for every $i = \Delta, \dotsc, 1$ , sequentially take ${\varepsilon }_i ={\varepsilon }_{i+1}\lambda _{i+1}$ and $\lambda _i = \lambda _{2.2}({\varepsilon }_i,\alpha )$ . Lastly, let $\lambda = \prod _{i \in [\Delta +1]} \lambda _i$ .

Fix any $r$ -colouring of (the edges of) $\Gamma$ and, for every $c \in [r]$ , let $\Gamma _c$ stand for the subgraph (in terms of edges) in colour $c$ . Note that $H$ has edge-chromatic number at most $\Delta +1$ . In other words, there exists a partition of the edges of $H$ into $H_1,\dotsc,H_{\Delta +1}$ such that each $H_i$ is a matching. We find the required collection $\{U_x\}_{x \in V(H)}$ by maintaining the following condition for every $i \in [\Delta +1]$ : for every $x \in V(H)$ , there exists a chain $V_x = U_x^0 \supseteq U_x^1 \supseteq U_x^2 \supseteq \dotsb \supseteq U_x^i$ such that

1. (i) $|U_x^j| = \lambda _j|U_x^{j-1}|$ for all $j \in [i]$ and

2. (ii) for every $xy \in \bigcup _{j \leqslant i} H_j$ , $(U_x^i, U_y^i)$ is $({\varepsilon }_i,\alpha p)$ -lower-regular in $\Gamma _c$ for some $c \in [r]$ .

Consequently, for $i = \Delta +1$ , we obtain sets $U_x \subseteq V_x$ , for every $x \in V(H)$ , of order $|U_x| = \left(\prod _{i \in [\Delta +1]} \lambda _i\right)\! n = \lambda n$ such that $(U_x,U_y)$ is $({\varepsilon }_{\Delta +1},\alpha p)$ -lower-regular and, thus, $({\varepsilon },\alpha p)$ -lower-regular for every $xy \in H$ . It remains to show that we can indeed do this.

Consider first $i = 1$ . For each $xy \in H_1$ , let $c \in [r]$ be the majority colour in $\Gamma [V_x,V_y]$ . As $e_{\Gamma _c}(V_x, V_y) \geqslant (1-\lambda )n^2p/r$ , we may apply Lemma 2.2 with ${\varepsilon }_1$ (as $\varepsilon$ ) and $\Gamma _c[V_x,V_y]$ (as $G$ ) to obtain sets $U_x^1, U_y^1$ with the desired properties. For every $x \in V(H)$ which is isolated in $H_1$ , we simply take an arbitrary subset $U_x^1 \subseteq V_x$ of order $\lambda _1|U_x^0|$ . Thus, the required condition holds for $i=1$ .

Suppose now that the condition holds for some $i \geqslant 1$ and let us show that it also holds for $i+1$ . As above, for every $xy \in H_{i+1}$ , let $c \in [r]$ be the majority colour in $\Gamma [V_x,V_y]$ . Since $\Gamma [V_x, V_y]$ is $(\lambda,p)$ -uniform and, by (i), $|U_x^i|, |U_y^i| \geqslant \lambda n$ , we have $e_\Gamma (U_x^i, U_y^i) = (1 \pm \lambda )|U_x^i||U_y^i|p$ and, hence,

\begin{equation*} (1-\lambda )|U_x^i||U_y^i|p/r \leqslant e_{\Gamma _c}(U_x^i,U_y^i) \leqslant (1+\lambda )|U_x^i||U_y^i|p. \end{equation*}

Lemma 2.2 applied to $\Gamma _c\!\left[U_x^i,U_y^i\right]$ with ${\varepsilon }_{i+1}$ (as $\varepsilon$ ) gives sets $U_x^{i+1} \subseteq U_x^i$ and $U_y^{i+1} \subseteq U_y^i$ of order

\begin{equation*} \left|U_x^{i+1}\right| = \lambda _{i+1}\left|U_x^i\right| \qquad \text {and} \qquad \left|U_y^{i+1}\right| = \lambda _{i+1}\left|U_y^i\right| \end{equation*}

for which $\left(U_x^{i+1},U_y^{i+1}\right)$ is $({\varepsilon }_{i+1},\alpha p)$ -lower-regular in $\Gamma _c$ . For every $x \in V(H)$ which is isolated in $H_{i+1}$ , we again take an arbitrary subset $U_x^{i+1} \subseteq U_x^i$ of order $\lambda _{i+1}|U_x^i|$ . Observe also that, for every $xz \in \bigcup _{j \leqslant i} H_j$ , since $(U_x^i,U_z^i)$ was $({\varepsilon }_i,\alpha p)$ -lower-regular in $\Gamma_{c^{\prime}}$ for some $c' \in [r]$ and $|U_x^{i+1}| = \lambda _{i+1}|U_x^i|$ , Lemma 2.1 and the fact that ${\varepsilon }_i/\lambda _{i+1} ={\varepsilon }_{i+1}$ imply that $\left(U_x^{i+1},U_z^{i+1}\right)$ is $({\varepsilon }_{i+1},\alpha p)$ -lower-regular in $\Gamma_{c^{\prime}}$ , as desired. This completes the proof.

We also need a variant of a result from our previous paper [Reference Conlon, Nenadov and Trujić7, Lemma 3.5] about regularity inheritance. While that result was stated for the usual (full) notion of regularity, we only need lower-regularity here, allowing us to save a factor of $(\log n)^{1/2}$ .

3. Proof of Theorem 1.1

Since it requires no additional work, we will actually prove the $r$ -colour analogue of Theorem 1.1. More precisely, we will show that for every integer $r \geqslant 2$ there exists a graph of order $n$ with $O(n^{5/4})$ edges for which every $r$ -colouring of the edges contains a monochromatic copy of the $\delta \sqrt{n} \times \delta \sqrt{n}$ grid for some $\delta \gt 0$ .

By a result of Haxell, Kohayakawa, and Łuczak [Reference Haxell, Kohayakawa and Łuczak14, Theorem 10], there exist constants $K, \Delta \gt 0$ , both depending only on $r$ , such that, for every sufficiently large $s \in{\mathbb{N}}$ , there is a graph $H$ on $Ks$ vertices with maximum degree at most $\Delta$ which has the property that every $r$ -colouring of its edges contains a monochromatic copy of $C_\ell$ , the cycle of length $\ell$ , for every $\log s \ll \ell \leqslant s$ . Let

\begin{equation*} \alpha = 1/(2r), \!\quad\! {\varepsilon } = \alpha/256, \!\quad\! {\varepsilon }' = {\varepsilon }'_{\!\!2.4}({\varepsilon }/9, \alpha ), \quad \lambda = \lambda _{2.3}(r, \Delta, {\varepsilon }'), \!\quad\! \text {and} \quad \delta = \min \{1/(4K), {\varepsilon }\lambda/4 \}. \end{equation*}

We show that the size-Ramsey number of the $\delta s \times \delta s$ grid is $O(s^{5/2})$ , which, for $s = \sqrt{n}$ , implies the desired statement.

Let $\Gamma$ be a graph obtained by replacing every vertex $x \in V(H)$ by an independent set $V_x$ of order $s$ and every edge $xy \in H$ by a bipartite graph between $V_x$ and $V_y$ in which each edge exists independently with probability $p = Cs^{-1/2}$ for some sufficiently large constant $C \gt 0$ . With high probability, $\Gamma$ has the following property:

1. (A1) $e_\Gamma (V'_{\!\!x}, V'_{\!\!y}) = (1\pm \lambda )|V'_{\!\!x}||V'_{\!\!y}| p$ for every $xy \in H$ and $V'_{\!\!x} \subseteq V_x$ and $V'_{\!\!y} \subseteq V_y$ with $|V'_{\!\!x}||V'_{\!\!y}|p \geqslant 100s/\lambda ^2$ .

This is a standard feature of random graphs and follows from the Chernoff bound together with an application of the union bound. In particular, it establishes that with high probability $\Gamma [V_x,V_y]$ is $(\lambda,p)$ -uniform for every $xy \in H$ and, therefore, $\Gamma$ has at most

\begin{equation*} Ks \cdot \Delta/2 \cdot (1+\lambda )s^2p = O(s^{5/2}) \end{equation*}

edges. Additionally, with high probability, $\Gamma$ is such that every $\Gamma [V_x \cup V_y \cup V_z]$ has the property of Lemma 2.4 (applied with ${\varepsilon }/9$ as $\varepsilon$ , $\lambda/3$ as $\lambda$ , and $3s$ as $n$ ) for every path $xyz$ of length two in $H$ .Footnote ² This again follows from the union bound, as there are $O(s)$ such paths in total and the conclusion of Lemma 2.4 holds with probability $1 - o(s^{-5})$ for every fixed path. We now fix an outcome of $\Gamma$ which satisfies all of these properties.

Consider some $r$ -colouring of the edges of $\Gamma$ and let $\varphi$ be the colouring of the edges of $H$ given by Lemma 2.3 (applied with $\varepsilon '$ as $\varepsilon$ ). By the choice of $H$ , this colouring contains a monochromatic copy of $C_{\delta s}$ , which, without loss of generality, we may assume has vertices $1,\dotsc,\delta s$ . Therefore, there is a colour $c \in [r]$ and sets $U_i$ of order $\tilde s = \lambda s$ in $\Gamma$ such that, for every $i \in [\delta s]$ , the pair $(U_i,U_{i+1})$ is $({\varepsilon }',\alpha p)$ -lower-regular in the subgraph of $\Gamma$ induced by colour $c$ , where we identify $\delta s + i$ with $i$ . Let $G$ be the graph induced by these sets whose edges are the edges of $\Gamma$ of colour $c$ . We will show that $G$ contains the $\delta s \times \delta s$ grid as a subgraph.

For every $i \in [\delta s]$ , let $B \subseteq U_i \cup U_{i+1} \cup U_{i+2}$ be the set given by Lemma 2.4 (which was applied with ${\varepsilon }/9$ as $\varepsilon$ , $\lambda/3$ as $\lambda$ , and $3s$ as $n$ ) on $\Gamma [U_i \cup U_{i+1} \cup U_{i+2}]$ , which is a set of ‘bad vertices’ for the pair $(U_{i+1}, U_{i+2})$ . As each $U_i$ is a part of three such applications, by the chosen properties of $\Gamma$ , for every $i \in [\delta s]$ there exists a set $B_i \subseteq U_i$ of order $|B_i| \leqslant{\varepsilon }\tilde s$ such that:

1. (B1) $(N_v, U_{i+2} \setminus B_{i+2})$ is $({\varepsilon }, \alpha p)$ -lower-regularFootnote ³ in $G$ for every $v \in U_i \setminus B_i$ and $N_v \subseteq N_G(v, U_{i+1})$ of order $\alpha \tilde sp/4$ and

2. (B2) $(N_v, N_u)$ is $({\varepsilon }, \alpha p)$ -lower-regular in $G$ for every $v \in U_i \setminus B_i$ , $u \in U_{i+1} \setminus B_{i+1}$ and $N_v \subseteq N_G(v, U_{i+1})$ , $N_u \subseteq N_G(u, U_{i+2})$ , each of order $\alpha \tilde sp/4$ .

Our plan is to embed the vertex $(i, j)$ of the $\delta s \times \delta s$ grid into $U_{i+j-1}$ . The next claim helps us achieve this.

Before proving the claim, we show how to complete the embedding of the grid assuming that it holds. We start by embedding the first row. Let $v_1 \in U_1 \setminus B_1$ be a vertex for which there is $S_2 \subseteq N_G(v_1, U_2 \setminus B_2)$ of order $\alpha \tilde sp/4$ such that $(S_2, U_3 \setminus B_3)$ is $({\varepsilon }, \alpha p)$ -lower-regular. As $(U_1 \setminus B_1, U_2 \setminus B_2)$ is $(2{\varepsilon }', \alpha p)$ -lower-regular, there are at least $(1-2{\varepsilon }')(1-{\varepsilon })\tilde s$ vertices $v \in U_1 \setminus B_1$ that satisfy

\begin{equation*} \deg _G(v, U_2 \setminus B_2) \geqslant (1-2{\varepsilon }')|U_2 \setminus B_2|\alpha p \geqslant \alpha \tilde sp/4, \end{equation*}

by our choice of constants. Thus, by property () almost any choice of $v_1 \in U_1 \setminus B_1$ will do. Sequentially, for every $i \geqslant 2$ , let $v_i \in S_i$ be a vertex for which there is $S_{i+1} \subseteq N_G(v_i, U_{i+1} \setminus B_{i+1})$ of order $\alpha \tilde sp/4$ and both $(S_{i+1}, U_{i+2} \setminus B_{i+2})$ and $(S_i, S_{i+1})$ are $({\varepsilon }, \alpha p)$ -lower-regular. This is possible as $(S_i, U_{i+1} \setminus B_{i+1})$ is $({\varepsilon }, \alpha p)$ -lower-regular and properties () and () hold. We continue until we have embedded the first row of the grid as $v_1,\dotsc,v_{\delta s}$ , with $v_i \in U_i$ for every $i \in [\delta s]$ .

Consider now sets $S_2, \dotsc, S_{\delta s}, S_1$ which we previously chose, where we note that $S_1$ was defined when we embedded $v_{\delta s}$ . In particular, $S_{1+j} \subseteq U_{1+j} \setminus B_{1+j}$ and $(S_{1+j}, S_{2+j})$ and $(S_{1+j}, U_{2+j} \setminus B_{2+j})$ are both $({\varepsilon }, \alpha p)$ -regular for every $j \in [\delta s]$ . Then, by setting $Q_{1+j} \;:\!=\; B_{1+j} \cup \{v_{1+j}\}$ and invoking Claim 3.1 with $i = 2$ , we can embed the second row of the grid as $u_1, \dotsc, u_{\delta s}$ , with $u_j \in S_{1+j}$ for every $j \in [\delta s]$ . By the conclusion of Claim 3.1 and a slight abuse of notation, there is a collection of sets $S_{2+j} \subseteq N_G(u_j, U_{2+j} \setminus Q_{2+j})$ for every $j \in [\delta s]$ , each of order $\alpha \tilde sp/4$ , which, by () and (), as $u_j \in U_{1+j} \setminus B_{1+j}$ and $u_{j+1} \in U_{1+j+1} \setminus B_{1+j+1}$ , are such that $(S_{2+j}, S_{2+j+1})$ and $(S_{2+j}, U_{2+j+1} \setminus B_{2+j+1})$ are $({\varepsilon }, \alpha p)$ -lower-regular.

The same process can now be repeated for any $i \geqslant 3$ (see Figure 1) by setting the sets $Q_{i+j-1} \subseteq U_{i+j-1}$ for every $j \in [\delta s]$ to be the union of $B_{i+j-1}$ and the vertices of the grid that were previously embedded into $U_{i+j-1}$ , that is, the images of the vertices $(1,i+j-1), (2,i+j-2), \dotsc, (i-1,j+1)$ . Since $|B_{i+j-1}| \leqslant{\varepsilon }\tilde s$ , $\delta \lt{\varepsilon }\lambda$ , and the lower-regularity conditions hold by () and (), we may apply Claim 3.1 to embed the $i$ th row. It only remains to prove this claim.

Proof of Claim 3.1. Without loss of generality, we may assume that all the $Q_{i+j-1}$ are of order $2{\varepsilon }\tilde s$ , as we can take arbitrary supersets if this is not the case. Let $S'_{\!\!i+j-1} \subseteq S_{i+j-1}$ be the set of all $v \in S_{i+j-1}$ with at least $\alpha \tilde sp/4$ neighbours in $U_{i+j} \setminus Q_{i+j}$ . On the one hand, as $(S_{i+j-1}, U_{i+j} \setminus B_{i+j})$ is $({\varepsilon },\alpha p)$ -lower-regular and, thus, there are fewer than ${\varepsilon }|S_{i+j-1}|$ vertices in $S_{i+j-1}$ with degree less than $\alpha \tilde sp/2$ in $U_{i+j} \setminus B_{i+j}$ , we have

\begin{equation*} e_G(S_{i+j-1} \setminus S'_{\!\!i+j-1}, Q_{i+j}) \geqslant \big (|S_{i+j-1} \setminus S'_{\!\!i+j-1}| - {\varepsilon }|S_{i+j-1}|\big ) \alpha \tilde sp/4. \end{equation*}

On the other hand, assuming $S_{i+j-1} \setminus S'_{\!\!i+j-1}$ is of order at least $\alpha \tilde sp/16$ and, hence,

\begin{equation*} |S_{i+j-1} \setminus S'_{\!\!i+j-1}||Q_{i+j}|p \geqslant \alpha \tilde sp/16 \cdot 2{\varepsilon }\tilde sp \geqslant 100s/\lambda ^2 \end{equation*}

for $C \gt 0$ sufficiently large, property () implies that

\begin{equation*} e_G(S_{i+j-1} \setminus S'_{\!\!i+j-1}, Q_{i+j}) \leqslant (1+\lambda ) 2{\varepsilon }\tilde s |S_{i+j-1} \setminus S'_{\!\!i+j-1}|p. \end{equation*}

Since ${\varepsilon } \lt \alpha/128$ , this is a contradiction. Therefore, there are sets $S'_{\!\!i+j-1} \subseteq S_{i+j-1}$ of order at least $|S_{i+j-1}| - \alpha \tilde sp/16$ for each $j \in [\delta s]$ such that every $v \in S'_{\!\!i+j-1}$ satisfies $|N_G(v, U_{i+j} \setminus Q_{i+j})| \geqslant \alpha \tilde sp/4$ .

Figure 1. A picture showing the first two rows of the grid already embedded (the thick black lines), the candidate sets for the third row (the grey blobs $S_3, S_4, \dotsc, S_{\delta s}, S_1, S_2$ ), and (in red) the path $v_1, v_2, \dotsc, v_{\delta s}$ given by Claim 3.1, together with the corresponding neighbourhoods $N_G(v_j, U_{i+j} \setminus Q_{i+j})$ (the red blobs).

We will now find a collection of sets $S''_{\!\!\!i+j-1} \subseteq S'_{\!\!i+j-1}$ of order at least $|S_{i+j-1}| - \alpha \tilde sp/8$ such that, for every $2 \leqslant j \leqslant \delta s$ , every $v \in S''_{\!\!\!i+j-2}$ has a non-empty $N_G(v,S''_{\!\!\!i+j-1})$ . First, choose $S''_{\!\!\!i+\delta s-1} \subseteq S'_{\!\!i+\delta s-1}$ of order $|S_{i+\delta s-1}| - \alpha \tilde sp/8$ arbitrarily, noting that such a set exists by the bound on $|S'_{\!\!i+\delta s-1}|$ . Having chosen $S''_{\!\!\!i+j-1}$ for some $2 \leqslant j \leqslant \delta s$ , we choose $S''_{\!\!\!i+j-2}$ as follows. Recall that $(S_{i+j-2},S_{i+j-1})$ is $({\varepsilon },\alpha p)$ -lower-regular and, thus, by Lemma 2.1 and the bounds on the orders of $S'_{\!\!i+j-2}$ and $S''_{\!\!\!i+j-1}$ , $(S'_{\!\!i+j-2},S''_{\!\!\!i+j-1})$ is $(2{\varepsilon },\alpha p)$ -lower-regular. It follows that there are at least $(1-2{\varepsilon })|S'_{\!\!i+j-2}| \geqslant |S_{i+j-2}| - \alpha \tilde sp/8$ vertices $v \in S'_{\!\!i+j-2}$ which satisfy

\begin{equation*} \deg _G(v, S''_{\!\!\!i+j-1}) \geqslant (1-2{\varepsilon })|S''_{\!\!\!i+j-1}|\alpha p \geqslant \alpha ^2 \tilde sp^2/16 \gt 0. \end{equation*}

We declare the set of such vertices to be $S''_{\!\!\!i+j-2}$ and continue on to the next index $j$ .

Starting with an arbitrary $v_1 \in S''_{\!\!\!i}$ and sequentially choosing $v_j \in N_G(v_{j-1}, S''_{\!\!\!i+j-1})$ now completes the proof.