# Hypergraphs with vanishing Tur\'an density in uniformly dense hypergraphs

## TL;DR

This paper characterizes specific 3-uniform hypergraphs that must appear in large, uniformly dense hypergraphs of positive density, extending classical extremal graph theory to a uniform density setting.

## Contribution

It provides a characterization of hypergraphs guaranteed to appear in large uniformly dense hypergraphs, advancing the understanding of Turán-type problems in this context.

## Key findings

- Identifies hypergraphs with guaranteed appearance in uniformly dense hypergraphs.
- Extends Erdős's classical results to uniform density conditions.
- Analyzes cases where induced subhypergraph densities vary with vertex set proportions.

## Abstract

P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We study related questions for $3$-uniform hypergraphs with the additional assumption that $H$ has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs $F$ that are guaranteed to appear in large uniformly dense hypergraphs $H$ of positive density. We also review the case when the density of the induced subhypergraphs of $H$ may depend on the proportion of the considered vertex sets.