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Status: Unchecked ×Keyword: clique-free graphs ×Clear all

2 claims from 1 paper

1. Mathematics › Limits and Structures in Graph Theory

   Induced Subgraphs of $$K_r$$-Free Graphs and the Erdős–Rogers Problem

   Gishboliner, Janzer and C · COMBINATORICA · 2025

   The paper bounds the largest F-free induced subgraph guaranteed inside K_r-free graphs, showing sharp thresholds in the exponent and a contrast with off-diagonal Ramsey numbers for graphs F of large minimum degree.

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   1. UncheckedFor any K_{r-1}-free graph F, every K_r-free graph on n vertices has a large F-free induced subgraph, of size at most about n^(1/2-ε) in the worst case.“Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there exists some $\varepsilon_F>0$ such that $f_{F,K_r}(n)=O(n^{1/2-\varepsilon_F})$.”
   2. UncheckedFor every r≥4 and ε>0, some K_{r-1}-free graph F has f_{F,K_r}(n) of order at least n^{1/2-ε}, so a general n^{1/2} barrier is nearly sharp.“Secondly, we show that for all $r\geq 4$ and $\varepsilon>0$, there exists a $K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=Ω(n^{1/2-\varepsilon})$.”