Blowup Ramsey numbers

TL;DR

This paper explores generalized bipartite Ramsey numbers involving graph blowups, establishing bounds and conjecturing exponential growth rates, with results applicable to random graphs above certain thresholds.

Contribution

It introduces bounds for blowup Ramsey numbers and proposes a conjecture on their exponential growth, supported by probabilistic evidence.

Findings

Established exponential lower and upper bounds for blowup Ramsey numbers.

Conjectured the bounds grow exponentially with the blowup parameter t.

Confirmed the conjecture for random graphs above specific thresholds.

Abstract

We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph $G$, denote the $t$-blowup of $G$ by $G[t]$. We say that $G$ is $r$-Ramsey for $H$, and write $G \stackrel{r}{\rightarrow} H$, if every $r$-colouring of the edges of $G$ has a monochromatic copy of $H$. We show that if $G \stackrel{r}{\rightarrow} H$, then for all $t$, there exists $n$ such that $G[n] \stackrel{r}{\rightarrow} H[t]$. In fact, we provide exponential lower and upper bounds for the minimum $n$ with $G[n] \stackrel{r}{\rightarrow} H[t]$, and conjecture an upper bound of the form $c^t$, where $c$ depends on $H$ and $r$, but not on $G$. We also show that this conjecture holds for $G(n,p)$ with high probability, above the threshold for the event $G(n,p) \stackrel{r}{\rightarrow} H$.