Eoin Long

Distinct degrees and homogenous sets

In this talk I will discuss some recent work examining the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph G and the maximal number of distinct degrees appearing in an induced subgraph of G, denoted respectively by hom (G) and f(G). Our main theorem improves estimates due to Bukh and Sudakov and to Narayanan and Tomon and shows that if G is an n-vertex graph with hom (G) at least n^{1/2} then f(G) > ( n / hom (G) )^{1 - o(1)}. The bound here is sharp up to the o(1)-term, and asymptotically solves a conjecture of Narayanan and Tomon. In particular, this implies that max { hom (G), f(G) } > n^{1/2 -o(1)} for any n-vertex graph G, which is also sharp. The relationship between these parameters is known to change when hom (G) < n^{1/2}. I hope to discuss the suspected relationship in this other region, along with supporting results. Joint work with Laurențiu Ploscaru.