  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
$\Omega_C(N^{1/2})$ distinct degrees. We improve this to $\Omega_C(N^{2/3})$,
which is tight up to the constant factor.
  We also show that any $N$-vertex graph with $N > (k-1)(n-1)$ and $n\geq
n_0(k) = \Omega (k^9)$ either contains a homogeneous set of order $n$ 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.
