Two remarks on the Burr-Erdos conjecture

TL;DR

This paper improves bounds on the Ramsey numbers of d-degenerate graphs and analyzes the Ramsey numbers of random graphs, showing they are linear in n for fixed d, with nearly tight bounds for bipartite cases.

Contribution

It provides a new upper bound for the Ramsey number of d-degenerate graphs and studies the typical Ramsey number behavior of random graphs.

Findings

r(H)  2^{c_d\u221a{\u2113 n}} n for d-degenerate graphs

Almost surely, G(n,d/n) has linear Ramsey number in n

Nearly tight bounds for Ramsey numbers of random bipartite graphs

Abstract

The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erd\H{o}s in 1975 conjectured that for each positive integer d there is a constant c_d such that r(H) \leq c_dn for every d-degenerate graph H on n vertices. We show that for such graphs r(H) \leq 2^{c_d\sqrt{\log n}}n, improving on an earlier bound of Kostochka and Sudakov. We also study Ramsey numbers of random graphs, showing that for d fixed, almost surely the random graph G(n,d/n) has Ramsey number linear in n. For random bipartite graphs, our proof gives nearly tight bounds.