Two problems in graph Ramsey theory

Abstract

We study two problems in graph Ramsey theory. In the early 1970s, Erdős and O'Neil considered a generalization of Ramsey numbers. Given integers n,k,s and t with n≥k≥s,t≥2, they asked for the least integer N=fk(n,s,t) such that in any red–blue coloring of the k-subsets of {1,2,…,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. Erdős and O'Neil found an exact formula for fk(n,s,t) when k≥s+t−1. In the arguably more interesting case where k=s+t−2, they showed [Formula presented] for sufficiently large n. Our main result closes the gap between these lower and upper bounds, determining the logarithm of fs+t−2(n,s,t) up to a multiplicative factor. Recently, Damásdi, 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 Kn for which any extension of this to an r-edge-coloring of Kn+1 would create a new monochromatic copy of Kk. We obtain essentially sharp bounds for this problem.