Induced Tur\'an problems and traces of hypergraphs

TL;DR

This paper explores the relationship between induced Berge subgraphs in hypergraphs and classical Turán problems, establishing connections between hypergraph edge extremal functions and graph clique counts.

Contribution

It introduces a new framework linking induced Berge hypergraph problems with traditional Turán graph extremal functions, providing new insights into hypergraph trace problems.

Findings

Established a strong relation between hypergraph extremal functions and graph clique counts.

Derived bounds connecting induced Berge hypergraph problems with classical Turán problems.

Provided new methods for analyzing hypergraph traces in the context of extremal combinatorics.

Abstract

Let $F$ be a graph. We say that a hypergraph $H$ contains an induced Berge $F$ if the vertices of $F$ can be embedded to $H$ (e.g., $V(F)\subseteq V(H)$) and there exists an injective mapping $f$ from the edges of $F$ to the hyperedges of $H$ such that $f(xy) \cap V(F) = \{x,y\}$ holds for each edge $xy$ of $F$. In other words, $H$ contains $F$ as a trace. Let $ex_{r}(n,B_{ind} F)$ denote the maximum number of edges in an $r$-uniform hypergraph with no induced Berge $F$. Let $ex(n,K_r, F)$ denote the maximum number of $K_r$'s in an $F$-free graph on $n$ vertices. We show that these two Tur\'an type functions are strongly related.