Distinct degrees in induced subgraphs

Abstract

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 Erdős, Faudree and Sós, 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 $n\ge n_0(k)=\Omega (k^9)$ 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 Turán graph, and confirms a conjecture of Narayanan and Tomon.

13 pages