Independent Sets in Hypergraphs
Jacques Verstraete, Chase Wilson

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 -vertex triangle-free graph of maximum degree contains an independent set of size at least . Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di proved that every -uniform -vertex ``uncrowded'' hypergraph of maximum degree has an independent set of size at least for some depending only on . 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.
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Taxonomy
TopicsAdvanced Graph Theory Research
