Distinct degrees and homogeneous sets

Abstract

In this paper we investigate 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 several earlier researchers and shows that if G is an n-vertex graph with hom⁡(G)≥n1/2 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)}≥n1/2−o(1) for any n-vertex graph G, which is also sharp.The above relationship between hom⁡(G) and f(G) breaks down in the regime where hom⁡(G)<n1/2. Our second result provides a sharp bound for distinct degrees in biased random graphs, i.e. on f(G(n,p)). We believe that the behaviour here determines the extremal relationship between hom⁡(G) and f(G) in this second regime.Our approach to lower bounding f(G) proceeds via a translation into an (almost) equivalent probabilistic problem, and it can be shown to be effective for arbitrary graphs. It may be of independent interest.