Graphs and Matroids | Jun Yan, Ramsey numbers of trees

Abstract: Let T be a tree with bipartition class sizes t_1>=t_2. Motivated by two simple lower bound constructions, Burr conjectured that the Ramsey number of T, denoted by R(T), is exactly max{t_1+2t_2,2t_1}-1. While this conjecture turns out to be false, all known counterexamples have large maximum degrees. In a joint work with Richard Montgomery and Matías Pavez-Signé, we show that there exists a constant c>0, such that Burr's conjecture does hold if T has maximum degree at most c(t_1+t_2). In particular, this determines the exact Ramsey numbers of a large family of trees. In this talk, I will go over some background on tree embeddings, and give an overview of our proof.