Subgraphs with large minimum $\ell$-degree in hypergraphs where almost all $\ell$-degrees are large

TL;DR

This paper proves that in nearly uniform hypergraphs with most $inom{n}{ ext{ell}}$ $ ext{ell}$-subsets having high degree, there exists a large subgraph with a high minimum $ ext{ell}$-degree, extending understanding of hypergraph degree conditions.

Contribution

It establishes the existence of large subgraphs with high minimum $ ext{ell}$-degree in hypergraphs where almost all $ ext{ell}$-subsets are highly connected, generalizing previous degree threshold results.

Findings

Existence of large subgraphs with high minimum $ ext{ell}$-degree

Hypergraph degree conditions imply dense substructures

Extension of degree threshold concepts in hypergraphs

Abstract

Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that $G$ contains a large subgraph with high minimum $\ell$-degree.