Two problems in graph Ramsey theory

TL;DR

This paper advances the understanding of two problems in graph Ramsey theory, providing precise bounds for a generalized Ramsey number and saturation problems, thereby closing previous gaps in the theoretical framework.

Contribution

It determines the asymptotic behavior of a generalized Ramsey number for the case k=s+t-2 and provides sharp bounds for saturation problems in edge-colored complete graphs.

Findings

Closed the gap in bounds for f_{s+t-2}(n,s,t)

Established sharp bounds for saturation problems in Ramsey theory

Extended understanding of generalized Ramsey numbers and saturation phenomena

Abstract

We study two problems in graph Ramsey theory. In the early 1970's, Erd\H{o}s and O'Neil considered a generalization of Ramsey numbers. Given integers $n,k,s$ and $t$ with $n \ge k \ge s,t \ge 2$, they asked for the least integer $N=f_k(n,s,t)$ such that in any red-blue coloring of the $k$-subsets of $\{1, 2,\ldots, N\}$, there is a set of size $n$ such that either each of its $s$-subsets is contained in some red $k$-subset, or each of its $t$-subsets is contained in some blue $k$-subset. Erd\H{o}s and O'Neil found an exact formula for $f_k(n,s,t)$ when $k\ge s+t-1$. In the arguably more interesting case where $k=s+t-2$, they showed $2^{-\binom{k}{2}}n<\log f_k(n,s,t)<2n$ for sufficiently large $n$. Our main result closes the gap between these lower and upper bounds, determining the logarithm of $f_{s+t-2}(n,s,t)$ up to a multiplicative factor. Recently, Dam\'asdi, Keszegh, Malec, Tompkins, Wang and Zamora initiated the investigation of saturation problems in Ramsey theory, wherein one seeks to minimize $n$ such that there exists an $r$-edge-coloring of $K_n$ for which any extension of this to an $r$-edge-coloring of $K_{n+1}$ would create a new monochromatic copy of $K_k$. We obtain essentially sharp bounds for this problem.