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