Improved bounds for the Erd\H{o}s-Rogers $(s,s+2)$-problem

TL;DR

This paper extends recent advances in the Erd ext{"o}s-Rogers function, providing new upper bounds for the size of $K_s$-free induced subgraphs in $K_{s+2}$-free graphs, approaching known lower bounds.

Contribution

It generalizes the recent result for $f_{2,4}(n)$ to all $s \\geq 2$, establishing near-optimal upper bounds for $f_{s,s+2}(n)$.

Findings

Proves $f_{s,s+2}(n) \\leq n^{(2s-3)/(4s-5)+o(1)}$ for all $s \\geq 2$

Improves previous bounds and approaches known lower bounds

Extends techniques from recent work on $r(4,k)$ to broader cases

Abstract

For $2\leq s<t$, the Erd\H{o}s-Rogers function $f_{s,t}(n)$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. There has been an extensive amount of work towards estimating this function, but until very recently only the case $t=s+1$ was well understood. A recent breakthrough of Mattheus and Verstra\"ete on the Ramsey number $r(4,k)$ states that $f_{2,4}(n)\leq n^{1/3+o(1)}$, which matches the known lower bound up to the $o(1)$ term. In this paper we build on their approach and generalize this result by proving that $f_{s,s+2}(n)\leq n^{\frac{2s-3}{4s-5}+o(1)}$ holds for every $s\geq 2$. This comes close to the best known lower bound, improves a substantial body of work and is the best that any construction of similar kind can give.