2.1 Notation

In this subsection, in preparation to give an overview of the proofs, we introduce and recall some notation. An $r$ -edge-coloured (or just an $r$ -coloured) graph is a graph together with a labelled partition of its edge-set into $r$ parts. We view the parts of this partition as being labelled by different colours. We say that one coloured graph $G$ contains another coloured graph $H$ if there exists an injection $\phi \colon V(H)\to V(G)$ such that $(a,b)\in E(H)$ implies that $(\phi (a),\phi (b))\in E(G)$ and $(\phi (a),\phi (b))$ has the same colour as $(a,b)$ .

We say that a coloured graph $G$ is a homogeneous $t$ -blow-up of another coloured graph $H$ if $V(G)$ can be partitioned into $t$ -sized sets $V_1,\cdots, V_{|V(H)|}$ , each $G[V_i]$ is a monochromatic clique (in some colour), and for each $(i,j)\in E(H)$ of colour $c$ , $G[V_i,V_j]$ is a complete bipartite graph where each edge gets colour $c$ .

An $r$ -totally coloured graph is a graph whose vertices as well as its edges are given an $r$ -colouring. A coloured graph $G$ is a $t$ -blow-up of a totally coloured graph $H$ if $G$ is a $t$ -blow-up of $H$ with respect to the edge-colouring of $H$ , and $G[V_i]$ is a monochromatic clique in the same colour as the vertex $i\in H$ .

If $G$ is $2$ -coloured, $\overline{G}$ is used to denote the coloured graph with the two colours interchanged.

We say that an $r$ -coloured $G$ is locally $\varepsilon$ -balanced if each vertex of $G$ is incident to at least $\varepsilon |V(G)|$ edges in each of the $r$ colours. We say that an $r$ -coloured graph $G$ is globally $\varepsilon$ -balanced if each colour class has size at least $\varepsilon \binom{n}{2}$ . Throughout the paper, $\epsilon$ is assumed to be a positive real number.

For the convenience of the reader, in the following paragraph we collect every coloured and totally coloured graph we make reference to throughout the paper. For further clarity, in Figure 2, we provide an illustration of each of these graphs.

Let $P_1$ be the totally coloured $K_2$ with vertices receiving colour red, and the unique edge receiving colour blue. Note that the $t$ -blow-up of $P_1$ is a blue induced $K_{t,t}$ . Let $P_2$ be the totally coloured $K_2$ with one vertex receiving colour red, the other receiving colour blue, and the edge receiving colour red. Let $P_3$ be the totally coloured $K_4$ with edges $(1,2),(2,3), (3,4)$ and vertices $2$ and $3$ coloured blue, and all other edges and vertices coloured red. Note that a $t$ -blow-up of $P_3$ is locally $1/4$ -balanced. Moreover, let $P_{3}^{\circ }$ be the (not totally) coloured graph obtained from $P_3$ by discarding the vertex colours. Let $C_4$ be a $2$ -edge-coloured $K_4$ with the red edges forming a $4$ -cycle. Let $M_1$ be the properly $2$ -edge-coloured $K_{2,2}$ .

With our notation, the aforementioned theorems can be stated as follows.

Note that $\overline{P_3}$ is not included in the above list as $\overline{P_3}=P_3$ .

2.2 Proof overview

Although our proofs are short, the method is quite different from the other papers in the area. In this section, we aim to motivate our approach and give heuristic explanations for why Theorems 3 and 4 are true, at least qualitatively (i.e. with some finite $n=n(t,\varepsilon )$ ). We begin with a brief discussion of Theorem 1.1 in order to compare the different approaches. It is rather straightforward to obtain a proof of Theorem 1.1 if one is only concerned with obtaining $t$ -blow-ups where $t$ tends to infinity with $n$ . Indeed, applying the well-known Kővari-Sós-Turán Theorem to both red and blue colour classes, we can obtain one red and one blue $K_{s,s}$ , say $T_1$ and $T_2$ , respectively, where $s\sim \log n$ (we can ensure that $T_1$ and $T_2$ are vertex-disjoint). Apply Ramsey’s theorem to the four parts coming from $T_1$ and $T_2$ , and delete all vertices outside of the monochromatic cliques guaranteed by this application. Similarly, for each pair of the $4$ parts, in turn, apply the Kővari-Sós-Turán theorem to the majority colour class in the bipartite graph between the pair of parts to guarantee a monochromatic complete bipartite subgraph, at each iteration reducing the size of the parts logarithmically. This produces a $t$ -blow-up of a totally coloured $K_4$ (where $t\to \infty$ as $n\to \infty$ ) where we do not have precise control over the colouring; however, we know that the colouring is not monochromatic (there exists at least one blue and one red edge). It is easy to see that this implies that the totally coloured $K_4$ must contain one of $P_1$ , $P_2$ , $\overline{P_1}$ , $\overline{P_2}$ , as desired.

In [Reference Fox and Sudakov9], the dependent random choice method (see, e.g. [Reference Fox and Sudakov10]) is employed to obtain optimal bounds for Theorem 1.1, eliminating the need for nested applications of the Theorems of Ramsey and Kővari-Sós-Turán. However, the argument still boils down to finding a blow-up of a totally coloured graph which is not monochromatic, thereby producing a structure which necessarily contains one of the desired patterns. To prove Theorems 1.3 and 1.4, we find blow-ups of totally coloured graphs where each vertex is adjacent to edges of both colours. The argument given in the previous paragraph is too weak to achieve this, since it does not make use of the assumption that our colouring is locally balanced. The dependent random choice is quantitatively much stronger, but does not give any additional structural information.

Hence, we need an argument which will make use of the global structure of the colouring. Specifically, our main tool (Lemma 2.4), a strengthening of a theorem due to Nikiforov [Reference Nikiforov17], reduces the problem to finding certain small subgraphs appearing in $G$ with a positive density. Using Lemma 2.4, the statements of Theorem 1.3 and 1.4 reduce to the following two statements, respectively.

A natural tool for proving statements such as Proposition 2.1 and 2.2 is the Szemerédi Regularity Lemma and closely related removal lemmas (see, e.g. [Reference Conlon and Fox3]). For the sake of obtaining better bounds and more cogent proofs, we do not use the regularity method. However, the regularity method does provide a shorter proof of a quantitatively weaker version of Proposition 2.1, and thus Theorem 1.3. Indeed, using the Regularity Lemma, it is not hard to show that the vertex set of a $2$ -coloured $K_n$ with a vanishing density of both $C_4$ and $\overline{C}_4$ can essentially (i.e. after recolouring $o(n^2)$ edges), be partitioned into a red and a blue clique. Such a $2$ -coloured $K_n$ can at best be locally $(1/4+o(1))$ -balanced, by an easy optimisation argument (see Proposition 3.5), implying Proposition 2.1.

Proposition 2.2 is, however, more subtle, since even finding a single copy of the desired small subgraphs is a non-obvious extremal problem. To give some intuition on why the statement holds, let us prove an idealised version of it.

To clarify the connection between Proposition 2.3 and Proposition 2.2, notice that completing a colouring of an alternating red-blue 4-cycle to an edge-colouring of $K_4$ yields a $C_4, \overline{C_4}$ or a $P_{3}^{\circ }$ .

Finally, we state our main technical tool, which allows us to deduce Theorems 1.3 and 1.4 from the aforementioned Propositions. Nikiforov [Reference Nikiforov17] showed that any graph with a positive density of $K_\ell$ -copies contains a large complete $\ell$ -partite subgraph. We strengthen this statement to find homogeneous blow-ups in $r$ -coloured graphs. Recall that our definition of a homogeneous blow-up in a coloured graph requires that the parts of the blow-up induce monochromatic cliques, but the colours of the cliques are not specified.

Lemma 2.4 is proved in Section 3.2. We remark that the Lemma also gives a short proof of Theorem 1.1 with the stronger hypothesis that $n\geq 2^{100t/\varepsilon }$ . Indeed, an easy counting argument implies that any globally $\varepsilon$ -balanced 2-colouring of $K_n$ contains $\Omega (\varepsilon n^3)$ properly coloured two-edge paths (i.e. $K_{1, 2}$ -copies), so Lemma 2.4 yields a homogeneous $t$ -blow-up of the non-monochromatic edge-colouring of $K_{1,2}$ . A simple case distinction then gives a blow-up of $P_1$ , $P_2$ or one of their complements. A similar argument gives a concise proof (with a weaker dependence between $n$ and $\varepsilon$ ) of Theorem 1.4 from [Reference Bowen, Lamaison and Müyesser1], which is a generalisation of Theorem 1.1 to an arbitrary number of colours, originally proved via nested applications of the dependent random choice method.

3.1 Proofs of the main theorems

We now show how the main theorems follow easily from the two propositions on subgraph counts and Lemma 2.4.

3.2 Finding homogeneous blow-ups of small patterns

In this subsection, we prove Lemma 2.4, a variant of a result due to Nikiforov [Reference Nikiforov17]. The main ingredient is Lemma 3.1, which is about dense $\ell$ -uniform hypergraphs. Before stating and proving Lemma 3.1, we give the following definitions which are central due to the inductive nature of the proof.

Let $\mathcal{H}$ be an $\ell$ -partite $\ell$ -uniform hypergraph $\mathcal{H}$ on parts $V_1, \ldots, V_\ell$ . We usually specify the members of $E({\mathcal{H}})$ as $\ell$ -tuples $(v_1, \ldots v_\ell )$ (where $v_i\in V_i$ ), but we also abuse notation by saying that $(v_1, \ldots, v_\ell )$ contains $v_1$ , or by writing $(v_1, \ldots v_\ell ) = (v_1, \ldots, v_{\ell -1})+v_\ell$ . Let $K_2({\mathcal{H}})$ be the vertex pairs contained in edges of $\mathcal{H}$ . Moreover, we write $\partial{\mathcal{H}}$ for the collection of $(\ell -1)$ -tuples $(v_1, \ldots, v_{\ell -1})$ which are contained in some edge $(v_1, \ldots, v_\ell )$ of $\mathcal{H}$ , with $v_i \in V_i$ for $i \in [\ell ]$ . We emphasise that this notation is not standard since $\partial{\mathcal{H}}$ only contains $(\ell -1)$ -tuples from $V_1 \times \dots \times V_{\ell -1}$ . Given a graph $G$ isomorphic to a $K_{\ell }(m)$ (the complete $\ell$ -partite graph with vertex parts of size $m$ ), we say that $\mathcal{H}$ covers $G$ if $E(G) \subset K_2({\mathcal{H}})$ and there are $m$ disjoint $S \in E({\mathcal{H}})$ with $S \subset V(G)$ (in other words, $\mathcal{H}$ contains a matching of size $m$ on $V(G)$ ).

To prove Lemma 3.1, we need the following lemma for finding unbalanced complete bipartite graphs, which can be proved using the classical double counting argument of Kővari, Sós and Turán [Reference Kővari, Sós and Turán14].

We also need a standard upper bound on the $r$ -colour Ramsey number due to Erdős and Szekeres [Reference Erdős and Szekeres6]. Namely, any $r$ -colouring of $K_n$ contains a monochromatic clique on $\frac{\log n}{r \log r}$ vertices. We emphasise that we are mostly concerned with the case $r =2$ . We can now proceed with the proof.

Proof of Lemma 3.1. Assume that $c \leq (r \log r)^{-1}$ , since the statement for larger values of $c$ then follows. We prove the statement by induction on $\ell$ . The case $\ell =1$ follows from Ramsey’s Theorem – a colouring of $K_{cn}$ contains a monochromatic clique of size $ (r \log r)^{-1} \log (cn) \geq (2 r \log r)^{-1} \log (n) \geq \frac c2 \log n$ . Assume that the statement holds for $\ell -1$ , and let $\mathcal{H}$ be as in the statement.

For a hypergraph $\mathcal{L}$ and $R= (v_1, \ldots, v_{\ell -1})$ , we write $d_{\mathcal{L}}(R)$ for the number of edges of $\mathcal{L}$ containing $R$ . A standard deletion argument shows that there is $\mathcal{L} \subset{\mathcal{H}}$ with $|\mathcal{L}| \geq (\ell -1)cn^\ell$ such that

(1) \begin{equation} \text{for any } R \in V_1 \times \dots \times V_{\ell -1}, \quad d_{\mathcal{L}}(R) \geq cn \,\,\text{or}\,\, d_{\mathcal{L}}(R)=0; \end{equation}

indeed, one can iteratively remove all edges containing $R$ for any $R$ violating (1), removing at most $cn \cdot n^{\ell -1}$ edges in total.

We have $|\partial \mathcal{L}| \geq |\mathcal{L}|/ n \geq (\ell -1) cn^{\ell -1}$ , since each $R \in \partial \mathcal{L}$ is contained in at most $n$ edges of $\mathcal{L}$ . Applying the induction hypothesis to $\partial \mathcal{L}$ , we obtain sets $U_1, \ldots U_{\ell -1}\subseteq V_1,\ldots, V_{\ell -1}$ with $|U_i| = m = \left (\frac c2 \right )^{\ell -1} \log n$ such that $\partial \mathcal{L}$ covers the complete $(\ell -1)$ -partite graph on $U_1 \cup \ldots \cup U_{\ell -1}$ , and $\varphi [U_i]$ is constant for each $i\in [\ell -1]$ .

Let $\mathcal{A}$ be a set of $m$ disjoint $(\ell -1)$ -tuples in $\partial \mathcal{L}$ , which exists by the definition of a covering. The graph structure of $K_2(\mathcal{L})$ will now be used by noticing that it suffices to find a large subset of vertices $T \subset V_\ell$ such that $R+v_\ell \in \mathcal{L}$ for all $R \in \mathcal{A}$ and $v_\ell \in T$ . To this end, consider the bipartite graph $F$ with parts $\mathcal{A}$ and $V_\ell$ such that $(R, v_\ell ) \in F$ whenever $R+ v_\ell$ lies in $\mathcal{L}$ . Since $d_{\mathcal{L}}(R) \gt cn$ for all $R \in \mathcal{A} \subset \partial \mathcal{L}$ , we have that $e(F) \geq cmn$ .

We can apply Lemma 3.2 to $F$ with $\alpha =(c/2)^\ell$ . We also set $s\;:\!=\; \left (\frac c2 \right )^\ell \log n =\alpha \log n \leq \frac c2 m + 1$ and $t\;:\!=\; n^{1-2^{-\ell }c^{\ell -1}} = n^{1-\alpha/ c}$ . It follows that $F$ contains a complete bipartite graph with parts $\mathcal{A}'$ and $T$ such that $|\mathcal{A}'| = s$ and $|T| = t$ . Let $G = K_2(\mathcal{L})$ . Let $S_1, \ldots, S_{\ell -1}$ be the vertex sets of the edges of $\mathcal{A}'$ , and recall that they induce a $K_{\ell -1}(s)$ in $ G$ since $S_i \subset U_i$ . Moreover, we claim that $w v_\ell$ is in $G$ for any $v_\ell \in T$ and $w \in S_i$ with $i \in [\ell -1]$ . This follows from the fact that there is an $R \in \mathcal{A}'$ containing $w$ , and $R+v_\ell \in \mathcal{L}$ .

Finally, by Ramsey’s theorem, there is a subset $S_\ell \subset T$ with $|S_\ell | = s = \left (\frac c2 \right )^\ell \log n \leq \frac{1}{r\log r} \log |T|$ on which $\varphi$ is constant. Recalling that $\varphi [S_i]$ is constant for $i \in [\ell -1]$ by induction hypothesis, we obtain our desired sets $S_1, \ldots, S_\ell$ . To verify that $\mathcal{L}$ covers $S_1, \ldots, S_\ell$ , note that the edges of $\mathcal{A}'$ with the vertices of $S_\ell$ (in an arbitrary order) form a matching of size $s$ in $\mathcal{L}$ .

We now give the proof of Lemma 2.4.

3.3 Small subgraphs in locally balanced colourings

We now prove Proposition 2.1, which follows immediately from Lemma 3.4 and Proposition 3.5. As discussed above, in Lemma 3.4 we actually describe the structure of colourings with a vanishing density of $C_4$ and $\overline{C_4}$ . We call a $2$ -coloured $K_n$ split if its vertex set can be partitioned into a red clique and a blue clique. We call two $2$ -coloured $K_n$ $\delta$ -close if the first can be made isomorphic to the second after flipping the colours of at most $\delta n^2$ many edges. The following result can be thought of as a substitute for the regularity method which is sufficient for our purposes.

We now use the above lemma to show that a $2$ -coloured graph with a vanishing density of $C_4$ and $\overline{C_4}$ is $o(1)$ -close to being split.

Proof. Suppose that $(1)$ does not hold. Then, by Lemma 3.3 applied with $\delta ^{5}$ in place of $\varepsilon$ , we have that there is an equitable partition of $G$ into at most

\begin{equation*}C=4/(\delta ^5 2^{-4c(\log (1/\delta ^5))^2})\end{equation*}

parts, each of which has either red or blue density above $1-\delta ^5$ . Let us refer to the parts with high red density as red parts, and label them by $V_1,\ldots, V_\ell$ .

Proof. We claim that $G[V_1 \cup \dots \cup V_\ell ]$ contains at most $\delta n^2/2$ blue edges. If $\ell = 1$ , this is trivial, and otherwise it follows from the following claim.

Claim 1. Let $V_i$ and $V_j$ be two parts with density $\geq 1-\delta ^5$ in red. Then the bipartite graph $G[V_i, V_j]$ has red density at least $1-\delta$ .

Suppose that the blue density between $V_i$ and $V_j$ is at least $\delta$ . It follows that the blue subgraph of $G[V_i\cup V_j]$ must contain at least $2^{-10} \delta ^4 (n/C)^4$ copies of a cycle on $4$ vertices (see, for example, Theorem 1.9(iv) from [Reference Nagy16]). At most $2\delta ^5 (n/C)^4$ such $4$ -cycles can contain a blue edge from $G[V_i]$ or $G[V_j]$ , by the density assumption. The remaining $4$ -cycles must correspond to a copy of $\overline{C_4}$ in $G[V_i\cup V_j]$ . Note that for some absolute constant $C_2$ , we have that $\delta ^4 (n/C)^4\geq 2^{-C_2\log (1/\delta )^8} n^4$ , so as we assumed that $(1)$ does not hold, we conclude that $G[V_i, V_j]$ has red density at least $1-\delta$ .

The same argument implies that the union of blue parts contains at most $\delta n^2/ 2$ blue edges. Hence, combining the red parts as well as the blue parts gives a partition certifying that $G$ is $\delta$ -close to a split graph, as required.

We now show that graphs which are $\delta$ -close to being split cannot be $(1/4+2\delta )$ -balanced. When combined with Lemma 3.4, this immediately implies Proposition 2.1.

Now we move on to Proposition 2.2. Recall that $M_1$ denotes a properly $2$ -edge-coloured $K_{2,2}$ . The following lemma is a robust version of Proposition 2.3. For a graph $G$ , let $\delta _G(S)$ denote the minimum degree (in $G$ ) of a vertex in $S \subset V(G)$ .

Proposition 2.2 follows easily from the previous result.