Distinct degrees in induced subgraphs

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and Sudakov, solving a conjecture of Erd\H{o}s, Faudree and S'os, shows that any $C$-Ramsey $N$-vertex graph contains an induced subgraph with ${\mathrm{\Omega }}_C(N^{1\mathrm{/}2})$ distinct degrees. We improve this to ${\mathrm{\Omega }}_C(N^{2\mathrm{/}3})$, which is tight up to the constant factor. We also show that any $N$-vertex graph with $N>(k-1)(n-1)$ and $n\geq n_0(k) = \Omega (k^9)$eithercontainsahomogeneoussetoforder$n$ or an induced subgraph with $k$ distinct degrees. The lower bound on $N$ here is sharp, as shown by an appropriate Tur'an graph, and confirms a conjecture of Narayanan and Tomon.