Induced Subgraphs with Many Distinct Degrees

Let $\hom(G)$ denote the size of the largest clique or independent set of a graph $G$. In 2007, Bukh and Sudakov proved that every $n$-vertex graph $G$ with $\hom(G) = O(\log n)$ contains an induced subgraph with $\Omega(n^{1/2})$ distinct degrees, and raised the question of deciding whether an analogous result holds for every $n$-vertex graph $G$ with $\hom(G) = O(n^\epsilon)$, where $\epsilon > 0$ is a fixed constant. Here, we answer their question in the affirmative and show that every graph $G$ on $n$ vertices contains an induced subgraph with $\Omega((n/\hom(G))^{1/2})$ distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed $k \in \mathbb{N}$, that if an $n$-vertex graph $G$ contains no induced subgraph with $k$ distinct degrees, then $\hom(G) \ge n/(k-1)-o(n)$; this bound is essentially best-possible.