Generalized Erdős-Rogers problems for hypergraphs
Xiaoyu He, Jiaxi Nie
TL;DR
This work studies the Erdős–Rogers-type function $f_{F,G}(n)$ for $r$-uniform hypergraphs, which captures how large an $F$-free induced subgraph one can guarantee inside every $G$-free host on $n$ vertices. It proves a polynomial lower bound whenever $G$ is contained in an $F$-iterated blowup, extending known graph results to hypergraphs via a supersaturation framework. It also derives polylogarithmic upper bounds when $G$ is $2$-tightly connected and not homomorphic to $F$, quantified by $\a_F=\max_{ eq ulle F' subseteq \d_2 F} rac{e(F')+1}{v(F')-1}$, and strengthens these results using the notion of $k$-shadow-homomorphisms to obtain $f^r_{F,G}(n) le c(\log n)^{1/(k-1)}$ under suitable non-homomorphism conditions. Collectively, the paper unifies and extends Erdős–Rogers-type bounds for hypergraphs, including corollaries for clique-like and simplex-like structures, and highlights several open directions in characterizing when the Erdős–Rogers function is polynomial versus polylogarithmic in $n$.
Abstract
Given $r$-uniform hypergraphs $G$ and $F$ and an integer $n$, let $f_{F,G}(n)$ be the maximum $m$ such that every $n$-vertex $G$-free $r$-graph has an $F$-free induced subgraph on $m$ vertices. We show that $f_{F,G}(n)$ is polynomial in $n$ when $G$ is a subgraph of an iterated blowup of $F$. As a partial converse, we show that if $G$ is not a subgraph of an $F$-iterated blowup and is $2$-tightly connected, then $f_{F,G}(n)$ is at most polylogarithmic in $n$. Our bounds generalize previous results of Dudek and Mubayi for the case when $F$ and $G$ are complete.