Three early problems on size Ramsey numbers

TL;DR

This paper investigates the size Ramsey numbers for specific graph families, resolving questions for book and starburst graphs, and providing significant progress on bipartite graphs, advancing understanding of these fundamental combinatorial parameters.

Contribution

The paper completely resolves the size Ramsey number questions for book and starburst graphs and advances the understanding of bipartite graphs for large parameters.

Findings

Resolved size Ramsey numbers for book and starburst graphs.

Determined size Ramsey number of $K_{s,t}$ up to a constant factor for large t.

Made substantial progress on bipartite graphs with $t = oldsymbol{ ext{Omega}}(s ext{log } s)$.

Abstract

The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of $H$ in every two-coloring of $E(G)$. The size Ramsey number was introduced by Erd\H{o}s, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex of a clique). In this paper, we completely resolve the latter two questions and make substantial progress on the first by determining the size Ramsey number of $K_{s,t}$ up to a constant factor for all $t = \Omega(s\log s)$.