Distinct degrees in induced subgraphs

TL;DR

This paper improves bounds on the number of distinct degrees in induced subgraphs of Ramsey graphs and establishes conditions for the existence of induced subgraphs with a specified number of distinct degrees.

Contribution

It enhances previous results by proving tighter bounds on the number of distinct degrees and confirms a conjecture regarding induced subgraphs with multiple degrees.

Findings

Improved the lower bound to _C(N^{2/3}) for distinct degrees in Ramsey graphs.

Established conditions under which graphs contain induced subgraphs with k distinct degrees.

Confirmed the sharpness of bounds using Ture1n graphs and proved a conjecture of Narayanan and Tomon.

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 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.