Distinct degrees and homogeneous sets II

TL;DR

This paper establishes the asymptotic relationship between the size of the largest homogeneous set and the number of distinct degrees in an induced subgraph of an n-vertex graph, confirming a conjecture and completing the extremal characterization.

Contribution

The authors prove a new lower bound on the number of distinct degrees in graphs with bounded homogeneous set size, confirming a previous conjecture and completing the extremal relationship between these parameters.

Findings

Proved that graphs with small homogeneous sets have many distinct degrees.

Confirmed the conjecture from previous work for the (G) regime.

Established tight asymptotic bounds for all homogeneous set sizes.

Abstract

Given an $n$-vertex graph $G$, let $\hom (G)$ denote the size of a largest homogeneous set in $G$ and let $f(G)$ denote the maximal number of distinct degrees appearing in an induced subgraph of $G$. The relationship between these parameters has been well studied by several researchers over the last 40 years, beginning with Erd\H{o}s, Faudree and S\'os in the Ramsey regime when $\hom (G) = O(\log n)$. Our main result here proves that any $n$-vertex graph $G$ with $\hom (G) \leq n^{1/2}$ satisfies \begin{align*} f(G) \geq \sqrt[3]{\frac {n^2}{\hom (G)} } \cdot n^{-o(1)}. \end{align*} This confirms a conjecture of the authors from a previous work, in which we addressed the $\hom (G) \geq n^{1/2}$ regime. Together, these provide the complete extremal relationship between these parameters (asymptotically), showing that any $n$-vertex graph $G$ satisfies \begin{align*} \max \Big ( f(G) \cdot \hom (G), \sqrt {f(G) ^3 \cdot \hom (G) } \Big ) \geq n^{1-o(1)}. \end{align*} This relationship is tight (up to the $n^{-o(1)}$ term) for all possible values of $\hom (G)$, from $\Omega (\log n )$ to $n$, as demonstrated by appropriately generated Erd\H{o}s $-$ Renyi random graphs.