Improved bounds for the Erd\H{o}s-Rogers function

TL;DR

This paper improves upper bounds for the Erd ext{"o}s-Rogers function in certain parameter ranges, showing that the function grows slower than previously established, with exponents less than 1/2.

Contribution

It provides the first proof that for specific pairs (s,t), the Erd ext{"o}s-Rogers function grows at a rate with an exponent less than 1/2, refining previous bounds.

Findings

Established new upper bounds for f_{s,t} with exponents < 1/2

Extended understanding of the growth rate of the Erd ext{"o}s-Rogers function

Answered an open question by Dudek, Retter, and R"{o}dl

Abstract

The Erd\H{o}s-Rogers function $f_{s,t}$ measures how large a $K_s$-free induced subgraph there must be in a $K_t$-free graph on $n$ vertices. While good estimates for $f_{s,t}$ are known for some pairs $(s,t)$, notably when $t=s+1$, in general there are significant gaps between the best known upper and lower bounds. We improve the upper bounds when $s+2\leq t\leq 2s-1$. For each such pair we obtain for the first time a proof that $f_{s,t}\leq n^{\alpha_{s,t}+o(1)}$ with an exponent $\alpha_{s,t}<1/2$, answering a question of Dudek, Retter and R\"{o}dl.