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 Ω(n) distinct degrees, and raised the question of deciding whether an analogous result holds for every n-vertex graph G with hom(G) = O(n), where ε > 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 Ω((n/ hom(G))) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed k ∈ N, that if an n-vertex graph G contains no induced subgraph with k distinct degrees, then hom(G) ≥ n/(k − 1)− o(n); this bound is essentially best-possible.