Distinct degrees and homogeneous sets

TL;DR

This paper establishes a near-optimal relationship between the size of the largest homogeneous set and the number of distinct degrees in an induced subgraph, solving a conjecture and analyzing random graphs.

Contribution

It improves bounds relating homogeneous sets and distinct degrees, proving a sharp estimate that resolves a conjecture and extends to biased random graphs.

Findings

For large homogeneous sets, the number of distinct degrees is at least (n/hom(G))^{1-o(1)}.

The bound is sharp up to the o(1) term, confirming the conjecture asymptotically.

Provides sharp bounds for distinct degrees in biased random graphs.

Abstract

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.