Three early problems on size Ramsey numbers

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 Erdős, 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, and starburst graphs. 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 = Ω(s log s).

Ramsey multiplicity and the Turán coloring

“…This gives the first examples of uncommon graphs whose Ramsey multiplicity constants are known exactly. Natural examples of graphs H to which Theorem 1.2 applies are starbursts (obtained from K k by adding an equal number of pendant edges to each vertex of K k ; see [9]) and pineapples (obtained from K k by adding pendant edges to a single vertex; see e.g. [49]).…”