Tiling Edge-Coloured Graphs with Few Monochromatic Bounded-Degree Graphs

Abstract

We prove that for all integers $\Delta ,r \ge 2$, there is a constant $C = C(\Delta ,r) >0$ such that the following is true for every sequence ${\mathcal {F}}= \{F_1, F_2, \ldots \}$ of graphs with $v(F_n) = n$ and $\Delta (F_n) \le \Delta$, for each $n \in {\mathbb {N}}$. In every r-edge-coloured $K_n$, there is a collection of at most C monochromatic copies from ${\mathcal {F}}$ whose vertex-sets partition $V(K_n)$. This makes progress on a conjecture of Grinshpun and Sárközy.

Appendix

In this appendix, we shall prove the lemmas stated in Sect. 3 for which we could not find a proof in the literature. Their proofs however are standard and not difficult.

Proof of Lemma 3.2

Let $U_i = (V_i \setminus X_i) \cup Y_i$ for $i \in \{1,2\}$. We will show that $(U_1,U_2)$ is $(8\varepsilon ,d - 8\varepsilon ,\delta /2)$-super-regular. Let now $Z_i \subseteq U_i$ with $|Z_i| \ge 8 \varepsilon |U_i|$, and let $Z_i' = Z_i {\setminus } Y_i$ and $Z_i'' = Z_i \cap Y_i$ for $i \in \{1,2\}$. Note that we have

$$\begin{aligned} |Z_i|&\ge 8\varepsilon |U_i| \ge \varepsilon |V_i|, \end{aligned}$$

(A.1)

$$\begin{aligned} |Z_i''|&\le |Y_i| \le \varepsilon ^2 |V_i| \overset{(A.1)}{\le } \varepsilon |Z_i| \text { and} \end{aligned}$$

(A.2)

$$\begin{aligned} |Z_i'|&= |Z_i| - |Z_i''| \overset{(A.2)}{\ge } (1-\varepsilon )|Z_i| \end{aligned}$$

(A.3)

for both $i \in \{1,2\}$. We therefore have

$$\begin{aligned}e(Z_1,Z_2) \le e(Z_1',Z_2') + e(Z_1'',Z_2) + e(Z_1,Z_2'') \overset{(A.2)}{\le } e(Z_1',Z_2') + 2\varepsilon |Z_1||Z_2|, \end{aligned}$$

and thus

$$\begin{aligned}d(Z_1,Z_2) \le d(Z_1',Z_2') + 2\varepsilon . \end{aligned}$$

On the other hand, we have

$$\begin{aligned} d(Z_1,Z_2)&= \frac{e(Z_1,Z_2)}{|Z_1||Z_2|} \ge \frac{e(Z_1',Z_2')}{|Z_1'||Z_2'|} \cdot \frac{|Z_1'||Z_2'|}{|Z_1||Z_2|} \\&\overset{(A.3)}{\ge } d(Z'_1,Z'_2){(1-\varepsilon )}^2 \ge d(Z_1',Z_2') - 2\varepsilon \end{aligned}$$

and hence $d(Z_1,Z_2) = d(Z_1',Z_2') \pm 2\varepsilon$. Furthermore, by $\varepsilon$-regularity of $(V_1,V_2)$, we have $d(Z_1',Z_2') = d(V_1, V_2) \pm \varepsilon$ and we conclude

$$\begin{aligned}d(Z_1,Z_2) = d(V_1, V_2) \pm 3\varepsilon . \end{aligned}$$

This holds in particular for $Z_1 = U_1$ and $Z_2 = U_2$ and therefore the pair $(U_1,U_2)$ is $(8\varepsilon ,d-8\varepsilon ,0)$-super-regular. Let $u_1 \in U_1$ now. By assumption, we have $\deg (u_1,V_2) \ge \delta |V_2|$ and therefore

$$\begin{aligned} \deg (u_1,U_2) \ge \deg (u_1,V_2 \setminus X_2)&\ge (\delta - \varepsilon ^2) |V_2| \\&\ge (\delta - \varepsilon ^2)|U_2| \ge \delta /2 \cdot |U_2|. \end{aligned}$$

A similar statement is true for every $u_2 \in U_2$ finishing the proof. $\square$

The following consequence of the slicing lemma will be useful when we prove Lemmas 3.4,3.5.

Lemma A.1

Let k be a positive integer and $d,\varepsilon >0$ with $\varepsilon \le 1/(2k)$. If $Z = (V_1, \ldots , V_k)$ is an $\varepsilon$-regular k-cylinder and $d(V_i,V_j) \ge d$ for all $1 \le i < j \le k$, then there is some $\gamma \le k\varepsilon$ and sets ${{\tilde{V}}}_1 \subseteq V_1, \ldots , {{\tilde{V}}}_k \subseteq V_k$ with $|{\tilde{V}}_i| = \lceil (1-\gamma ) |V_i| \rceil$ for all $i \in [k]$ so that the k-cylinder $\tilde{Z} = ({{\tilde{V}}}_1, \ldots , {{\tilde{V}}}_k)$ is $(2\varepsilon ,d - k\varepsilon )$-super-regular.

Proof

For $i \not = j \in [k]$, let $A_{i,j}:= \{v \in V_i: \deg (v,V_j) < (d-\varepsilon )|V_j|\}$. By definition of $\varepsilon$-regularity, we have $\left| A_{i,j} \right| < \varepsilon |V_i|$ for every $i \not = j \in [k]$. For each $i \in [k]$, let $A_i=\bigcup _{j \in [k]\setminus \{i\}} A_{i,j}$. Clearly $|A_i| < (k-1)\varepsilon |V_i|$ for every $i \in [k]$, so we can add arbitrary vertices from $V_i \setminus A_i$ to $A_i$ until $|A_i| = \lfloor (k-1) \varepsilon |V_i| \rfloor$ for every $i \in [k]$. Let now ${{\tilde{V}}}_i = V_i {\setminus } {\tilde{A}}_i$ for every $i \in [k]$ and let ${{\tilde{Z}}} = ({{\tilde{V}}}_1,\ldots ,{{\tilde{V}}}_k)$. Observe that $|{\tilde{V}}_i| = \lceil (1 - \gamma )|V_i| \rceil$ for all $i \in [k]$, where $\gamma = (k-1)\varepsilon$. It follows from Lemma 3.1 and definition of $A_i$ that ${{\tilde{Z}}}$ is $(2\varepsilon , d-\varepsilon , d-k\varepsilon )$-super-regular. $\square$

Given k disjoint sets $V_1,\ldots ,V_k$, we call a cylinder $(U_1, \ldots , U_k)$ relatively balanced (w.r.t. $(V_1, \ldots , V_k)$) if there exists some $\gamma > 0$ so that $U_i \subseteq V_i$ with $|U_i| = \lfloor \gamma |V_i| \rfloor$ for every $i \in [k]$. We say that a partition ${\mathcal {K}}$ of $V_1\times \cdots \times V_k$ is cylindrical if each partition class is of the form $W_1 \times \cdots \times W_k$ (which we associate with the k-cylinder $Z=(W_1, \ldots , W_k)$) with $W_j \subseteq V_j$ for every $j \in [k]$. Finally, we say that ${\mathcal {K}}= \{Z_1, \ldots , Z_N\}$ is $\varepsilon$-regular if

• (i) ${\mathcal {K}}$ is a cylindrical partition of $V_1\times \cdots \times V_k$,

• (ii) each $Z_i$, $i \in [k]$, is a relatively balanced w.r.t. $(V_1,\ldots ,V_k)$, and

• (iii) all but $\varepsilon |V_1|\cdots |V_k|$ of the k-tuples $(v_1,\ldots ,v_k) \in V_1 \times \cdots \times V_k$ are in $\varepsilon$-regular cylinders.

For technical reasons, we will allow some of the sets $V_1,\ldots ,V_k$ to be empty. In this case $(A,\emptyset )$ is considered $\varepsilon$-regular for every set A and $\varepsilon >0$. If G is an r-edge-coloured graph and $i \in [r]$, we say that a cylinder ${\mathcal {K}}$ is $\varepsilon$-regular in colour i if is $\varepsilon$-regular in $G_i$ (the graph on V(G) with all edges of colour i).

In [7], Conlon and Fox used the weak regularity lemma of Duke, Lefmann and Rödl [9] to find a reasonably large balanced k-cylinder in a k-partite graph. In order to prove a coloured version of Conlon and Fox’s result, we will need the following coloured version of the weak regularity lemma of Duke, Lefmann and Rödl. Note that, like the weak regularity lemma of Frieze and Kannan [15], we get an exponential bound on the number of cylinders, in contrast to the much worse tower-type bound required by Szemerédi’s regularity lemma (see [14]).

Theorem A.2

(Duke-Lefmann-Rödl [9]) Let $0< \varepsilon < 1/2$, $k, r \in {\mathbb {N}}$ and let $\beta = \varepsilon ^{rk^2 \varepsilon ^{-5}}$. Let G be an r-edge-coloured k-partite graph with parts $V_1, \ldots ,V_k$. Then there exist some $N \le \beta ^{-k}$, sets $R_1 \subseteq V_1, \ldots , R_k \subseteq V_k$ with $\left| R_i \right| \le \beta ^{-1}$ and a partition ${\mathcal {K}}= \{Z_1, \ldots , Z_N\}$ of $(V_1 \setminus R_1) \times \cdots \times (V_k \setminus R_k)$ so that ${\mathcal {K}}$ is $\varepsilon$-regular in every colour and $V_i(Z_j) \ge \lfloor \beta |V_i| \rfloor$ for every $i \in [k]$ and $j \in [N]$.

Although the original statement of Duke, Lefmann and Rödl [9, Proposition 2.1] does not involve the colouring and assume that sets $V_1,\ldots ,V_k$ have the same size, their proof can be easily adapted to prove Theorem A.2.

We are now ready to prove Lemmas 3.4,3.5.

Proof of Lemma 3.4

Let $k,r \ge 2$, $0< \varepsilon < 1/(rk)$ and $\gamma = \varepsilon ^{r^{8rk}\varepsilon ^{-5}}$. Let $n \ge 1/\gamma$ and suppose we are given an r-edge coloured $K_n$. Let ${\tilde{k}} = r^{rk}$ and let $V_1, \ldots , V_{{\tilde{k}}} \subseteq [n]$ be disjoint sets of size $\lfloor n/{\tilde{k}} \rfloor$ and let G be the ${\tilde{k}}$-partite subgraph of $K_n$ induced by $V_1, \ldots , V_{{\tilde{k}}}$ (inheriting the colouring). Let $\tilde{\varepsilon } = \varepsilon /2$ and $\beta = {\tilde{\varepsilon }}^{r^{2rk+1} {\tilde{\varepsilon }}^{-5}}$. We apply Theorem A.2 to get some $N \le \beta ^{-{{\tilde{k}}}}$, sets $R_1 \subseteq V_1,\ldots , R_{{{\tilde{k}}}} \subseteq V_{{{\tilde{k}}}}$ each of which of size at most $\beta ^{-1}$ and a partition ${\mathcal {K}}= \{Z_1, \ldots ,Z_N\}$ of $(V_1 {\setminus } R_i) \times \cdots \times (V_{{{\tilde{k}}}} {\setminus } R_{{{\tilde{k}}}})$ which is $\tilde{\varepsilon }$-regular in every colour, and with $V_i(Z_j) \ge \lfloor \beta |V_i| \rfloor \ge 2 \gamma n$ for every $i \in [{{\tilde{k}}}]$ and $j \in [N]$. Note that one of the cylinders (say $Z_1$) must be $\tilde{\varepsilon }$-regular in every colour and, since $(V_1, \ldots , V_k)$ is balanced, so is $Z_1$. We consider now the complete graph with vertex-set $\{V_1(Z_1), \ldots , V_{{\tilde{k}}}(Z_1)\}$ and colour every edge $V_i(Z_1)V_j(Z_1)$, $1 \le i < j \le {\tilde{k}}$, with a colour $c \in [r]$ so that the density of the pair $(V_i(Z_1),V_j(Z_1))$ in colour c is at least 1/r. By Ramsey’s theorem [12, 25], there is a colour, say 1, and k parts (say $V_1(Z_1), \ldots , V_k(Z_1)$) so that the cylinder $(V_1(Z_1), \ldots , V_k(Z_1))$ is $(\tilde{\varepsilon },1/r,0)$-super-regular in colour 1. By Lemma A.1, there is an $(\varepsilon ,1/(2r))$-super-regular balanced subcylinder ${{\tilde{Z}}}_1$ with parts of size at least $\gamma n$. $\square$

Proof of Lemma 3.5

Let $k \ge 2$, and let $d,\varepsilon >0$ with $2k\varepsilon \le d \le 1$. Let $\gamma = \varepsilon ^{k^2\varepsilon ^{-12}}$ and let G be a k-partite graph with parts $V_1, \ldots , V_k$. Let $\tilde{\varepsilon } = \varepsilon /4$ and $\beta = \tilde{\varepsilon }^{k^2 \tilde{\varepsilon }^{-5}}$. We may assume that $|V_i| \ge 1/\gamma$ for every $i \in [k]$ (otherwise we set $U_i:= \emptyset$ for all $i \in [k]$ with $|V_i| < 1/\gamma$). In particular, we have $|V_i| \ge k/({\tilde{\varepsilon }} \beta )$ for all $i \in [k]$.

We apply Theorem A.2 (with $r = 1$) to get some $N \le \beta ^{-k}$, sets $R_1 \subseteq V_1,\ldots , R_k \subseteq V_k$, each of which of size at most $\beta ^{-1}$, and an $\tilde{\varepsilon }$-regular partition ${\mathcal {K}}= \{Z_1, \ldots ,Z_N\}$ of $(V_1 {\setminus } R_1) \times \cdots \times (V_k {\setminus } R_k)$ with $V_i(Z_j) \ge \lfloor \beta |V_i| \rfloor$ for every $i \in [k]$ and $j \in [N]$.

Note that the number of cliques of size k incident to $R = R_1 \cup \ldots \cup R_k$ is at most

$$\begin{aligned} \sum _{i=1}^k \beta ^{-1} \prod _{j \in [k] \setminus \{i\}} |V_j| \le \tilde{\varepsilon } |V_1| \cdots |V_k|. \end{aligned}$$

Furthermore, since ${\mathcal {K}}$ is $\tilde{\varepsilon }$-regular, there are at most $\tilde{\varepsilon } |V_1| \cdots |V_k|$ cliques of size k in G that belong to a cylinder of ${\mathcal {K}}$ that is not $\varepsilon$-regular. Suppose that each cylinder $Z \in {\mathcal {K}}$ has at most $(d-2\tilde{\varepsilon })|V_1(Z)| \cdots |V_k(Z)|$ cliques of size k. Then the number of k-cliques in G is at most

$$\begin{aligned} \tilde{\varepsilon } |V_1| \cdots |V_k| + \sum _{Z \in {\mathcal {K}}} (d-2\tilde{\varepsilon }) |V_1(Z)| \cdots |V_k(Z)| \le (d-\tilde{\varepsilon })|V_1| \cdots |V_k|, \end{aligned}$$

which contradicts our hypothesis over G. Therefore, there is a cylinder ${{\tilde{Z}}}$ in ${\mathcal {K}}$ that contains at least $(d-2\tilde{\varepsilon })|V_1({{\tilde{Z}}})| \cdots |V_k({{\tilde{Z}}})|$ cliques of size k. In particular, ${{\tilde{Z}}}$ is $(\tilde{\varepsilon },d-2{\tilde{\varepsilon }},0)$-super-regular and relatively balanced with parts of size at least $\lfloor \beta |V_i| \rfloor$. Finally, we apply Lemma A.1 (and possibly delete a single vertex from some parts) to get a relatively balanced $(\varepsilon ,d-(k+2)\tilde{\varepsilon })$-super-regular k-cylinder Z with parts of size at least $\tfrac{\beta }{2} |V_i| \ge \gamma |V_i|$. This completes the proof since $(k+2)\tilde{\varepsilon } \le k\varepsilon \le d/2$. $\square$