Two Ramsey problems in blowups of graphs

Abstract

Given graphs $G$ and $H$, we say $G\stackrel{r}{\to }H$ if every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Let $H\left[t\right]$ denote the $t$-blowup of $H$. The blowup Ramsey number $B(G\stackrel{r}{\to }H;t)$ is the minimum $n$ such that $G\left[n\right]\stackrel{r}{\to }H\left[t\right]$. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given $G$, $H$ and $r$ such that $G\stackrel{r}{\to }H$, there exist constants $a=a(G,H,r)$ and $b=b(H,r)$ such that for all $t\in N$, $B(G\stackrel{r}{\to }H;t)\le a{b}^t$. They conjectured that there exist some graphs $H$ for which the constant $a$ depending on $G$ is necessary. We prove this conjecture by showing that the statement is true in the case of $H$ being 3-chromatically connected, which in particular includes triangles. On the other hand, perhaps surprisingly, we show that for forests $F$, there exists an upper bound for $B(G\stackrel{r}{\to }F;t)$ which is independent of $G$.

Second, we show that for any $r,t\in N$, any sufficiently large $r$-edge coloured complete graph on $n$ vertices with $\Omega \left(n^{2-1/t}\right)$ edges in each colour contains a member from a certain finite family $F_t^r$ of $r$-edge coloured complete graphs. This answers a conjecture of Bowen, Hansberg, Montejano and Müyesser.