Extremal and Ramsey results on graph blowups

TL;DR

This paper investigates blowup Ramsey numbers, proving that the exponential growth rate's dependence on the graph G can be eliminated, and offers a new proof of a key theorem with improved bounds.

Contribution

It demonstrates that the exponential constant in blowup Ramsey numbers does not depend on G, refining previous bounds and providing a new proof of a related theorem.

Findings

Dependence on G in the exponential constant is unnecessary.

Established a new proof of Nikiforov's theorem with better quantitative bounds.

Confirmed the conjecture that some dependence on G is unavoidable.

Abstract

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponential constant is indeed unnecessary, but conjecture that some dependence on $G$ is unavoidable. An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.