Independent Sets in Hypergraphs

TL;DR

This paper extends Shearer's method from triangle-free graphs to a broader class of hypergraphs, providing a simpler proof for the existence of large independent sets in these structures.

Contribution

It generalizes Shearer's approach to uniform hypergraphs, confirming his conjecture and simplifying the proof of a key theorem for locally sparse hypergraphs.

Findings

Extended Shearer's method to hypergraphs

Provided a short proof for a generalized theorem

Confirmed the existence of large independent sets in hypergraphs

Abstract

A theorem of Shearer states that every $n$-vertex triangle-free graph of maximum degree $d \geq 2$ contains an independent set of size at least $(d\log d - d + 1)/(d - 1)^2 \cdot n$. Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di proved that every $(r + 1)$-uniform $n$-vertex ``uncrowded'' hypergraph of maximum degree $d \geq 1$ has an independent set of size at least $c_r(\log d)^{1/r}/d^{1/r} \cdot n$ for some $c_r > 0$ depending only on $r$. Shearer asked whether his method for triangle-free graphs could be extended to uniform hypergraphs. In this paper, we answer this in the affirmative, thereby giving a short proof of the theorem of Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di for a wider class of ``locally sparse'' hypergraphs.