Generalized Ramsey numbers at the linear and quadratic thresholds

TL;DR

This paper improves bounds on generalized Ramsey numbers at linear and quadratic thresholds by strengthening connections to extremal problems and applying recent combinatorial methods.

Contribution

It provides improved estimates for $f(n, p, q)$ at key thresholds by strengthening known extremal problem links and utilizing recent forbidden submatchings techniques.

Findings

Enhanced bounds for $f(n, p, q_{ ext{lin}})$ and $f(n, p, q_{ ext{quad}})$.

Strengthened connection between Ramsey numbers and extremal problems.

Application of recent forbidden submatchings method to Ramsey number bounds.

Abstract

The generalized Ramsey number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that every $p$-clique spans at least $q$ colors. Erd\H{o}s and Gy\'arf\'as showed that $f(n, p, q)$ grows linearly in $n$ when $p$ is fixed and $q=q_{\text{lin}}(p):=\binom p2-p+3$. Similarly they showed that $f(n, p, q)$ is quadratic in $n$ when $p$ is fixed and $q=q_{\text{quad}}(p):=\binom p2-\frac p2+2$. In this note we improve on the known estimates for $f(n, p, q_{\text{lin}})$ and $f(n, p, q_{\text{quad}})$. Our proofs involve establishing a significant strengthening of a previously known connection between $f(n, p, q)$ and another extremal problem first studied by Brown, Erd\H{o}s and S\'os, as well as building on some recent progress on this extremal problem by Delcourt and Postle and by Shangguan. Also, our upper bound on $f(n, p, q_{\text{lin}})$ follows from an application of the recent forbidden submatchings method of Delcourt and Postle.