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On generalised Kneser colourings

Carsten Lange
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Abstract

There are two possible definitions of the "s-disjoint r-uniform Kneser hypergraph'' of a set system T: The hyperedges are either r-sets or r-multisets. We point out that Ziegler's (combinatorial) lower bound on the chromatic number of an s-disjoint r-uniform Kneser hypergraph only holds if we consider r-multisets as hyperedges. We give a new proof of his result and show by example that a similar result does not hold if one considers r-sets as hyperedges. In case of r-sets as hyperedges and $s \geq 2$ the only known lower bounds are obtained from topological invariants of associated simplicial complexes if r is a prime or the power of prime. This is also true for arbitrary r-uniform hypergraphs with r-sets or r-multisets as hyperedges as long as r is a power of a prime.

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Carsten Lange. 2003-12-02. On generalised Kneser colourings. https://arxiv.org/abs/math/0312067

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