
TL;DR
This paper introduces a new container method for hypergraph independent sets, providing a unified framework that advances understanding in combinatorics, coloring, extremal graph theory, and solutions to linear equations.
Contribution
It develops a novel hypergraph container technique that improves bounds and simplifies proofs across various combinatorial and graph-theoretic problems.
Findings
Established small collections covering all independent sets in hypergraphs.
Improved bounds on list chromatic number for hypergraphs of given degree.
Provided new bounds on the number of solution-free sets and hypergraph structures.
Abstract
We develop a notion of containment for independent sets in hypergraphs. For every -uniform hypergraph , we find a relatively small collection of vertex subsets, such that every independent set of is contained within a member of , and no member of is large; the collection, which is in various respects optimal, reveals an underlying structure to the independent sets. The containers offer a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. With regard to colouring, it follows that simple -uniform hypergraphs of average degree have list chromatic number at least . For this improves a bound due to Alon and is tight. For , previous bounds were weak but the present inequality is close to optimal. In the context of extremal graph theory, it follows…
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