Distinct degrees and homogeneous sets

In this paper we investigate the extremal relationship between two well-studied graph parameters: order of largest homogeneous set in a G and maximal number distinct degrees appearing an induced subgraph G, denoted respectively by hom⁡(G) f(G). Our main theorem improves estimates due to several earlier researchers shows that if is n-vertex with hom⁡(G)≥n1/2 then f(G)≥(n/hom⁡(G))1−o(1). The bound here sharp up o(1)-term, asymptotically solves conjecture Narayanan Tomon. particular, implies max⁡{hom⁡(G),f(G)}≥n1/2−o(1) for any which also sharp. above f(G) breaks down regime where hom⁡(G)<n1/2. second result provides biased random graphs, i.e. on f(G(n,p)). We believe behaviour determines regime. approach lower bounding proceeds via translation into (almost) equivalent probabilistic problem, it can be shown effective arbitrary graphs. It may independent interest.