English

Three early problems on size Ramsey numbers

Combinatorics 2023-02-09 v2

Abstract

The size Ramsey number of a graph HH is defined as the minimum number of edges in a graph GG such that there is a monochromatic copy of HH in every two-coloring of E(G)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 Ks,tK_{s,t} up to a constant factor for all t=Ω(slogs)t = \Omega(s\log s).

Keywords

Cite

@article{arxiv.2111.05420,
  title  = {Three early problems on size Ramsey numbers},
  author = {David Conlon and Jacob Fox and Yuval Wigderson},
  journal= {arXiv preprint arXiv:2111.05420},
  year   = {2023}
}

Comments

23 pages

R2 v1 2026-06-24T07:33:01.125Z