Erd\H{o}s-Rogers functions for arbitrary pairs of graphs

TL;DR

This paper investigates Erd ext{o}s-Rogers functions for various pairs of graphs, establishing asymptotic bounds and constructions that reveal how large induced subgraphs avoiding certain subgraphs can be guaranteed within larger graphs.

Contribution

It provides new asymptotic results for Erd ext{o}s-Rogers functions for specific graph pairs and improves existing constructions related to the Brown-Erd ext{o}s-Sós problem.

Findings

For triangle-free graphs F, f_{F,K_3}(n) = n^{1/2 + o(1)}.

For F containing a cycle and K_4-free, f_{F,K_4}(n) > n^{1/3 + c_F + o(1)}.

General bounds for f_{F,G}(n) depending on whether G is contained in blowups of F.

Abstract

Let $f_{F,G}(n)$ be the largest size of an induced $F$-free subgraph that every $n$-vertex $G$-free graph is guaranteed to contain. We prove that for any triangle-free graph $F$, \[ f_{F,K_3}(n) = f_{K_2,K_3}(n)^{1 + o(1)} = n^{\frac{1}{2} + o(1)}.\] Along the way we give a slight improvement of a construction of Erd\H os-Frankl-R\"odl for the Brown-Erd\H os-S\'os $(3r-3,3)$-problem when $r$ is large. In contrast to our result for $K_3$, for any $K_4$-free graph $F$ containing a cycle, we prove there exists $c_F > 0$ such that $$f_{F,K_4}(n) > f_{K_2,K_4}(n)^{1 + c_F} = n^{\frac{1}{3}+c_F+o(1)}.$$ \iffalse We also observe that our earlier proof for $F=K_3$ generalizes to $f_{F,K_4}(n) = O(\sqrt{n}\log n)$ for all $F$ containing a cycle. \fi For every graph $G$, we prove that there exists $\varepsilon_G >0$ such that whenever $F$ is a non-empty graph such that $G$ is not contained in any blowup of $F$, then $f_{F,G}(n) = O(n^{1-\varepsilon_G})$. On the other hand, for graph $G$ that is not a clique, and every $\varepsilon>0$, we exhibit a $G$-free graph $F$ such that $f_{F,G}(n) = \Omega(n^{1-\varepsilon})$.