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Status: Unchecked ×Keyword: hypergraph colouring ×Clear all

2 claims from 1 paper

1. Mathematics › Limits and Structures in Graph Theory

   Hypergraph containers

   Saxton and Thomason · Inventiones mathematicae · 2015

   The paper builds a 'container' method that gives a small collection of vertex sets covering every independent set of a hypergraph, and applies it to colouring, extremal graph theory and linear equations.

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   1. Unchecked“For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member of $C$, and no member of $C$ is large; the collection, which is in various respects optimal…
   2. UncheckedSimple r-uniform hypergraphs of average degree d have list chromatic number at least (1/(r-1)^2 + o(1)) log_r d, according to the paper.“With regard to colouring, it follows that simple $r$-uniform hypergraphs of average degree $d$ have list chromatic number at least $(1/(r-1)^2 + o(1)) \log_r d$.”