On the chromatic number of Kneser hypergraphs

Matoušek, Jiřı́

doi:10.1090/s0002-9939-02-06371-2

Cited by 10 publications

(7 citation statements)

References 12 publications

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“…Since then several different proofs have been appeared. One of the two main ways to prove the lower bound uses Borsuk-Ulam theorem (or its analogues), see [22,18,19,20] and the other computes the connectedness of a corresponding complex, see [17,3,15]. Recall that for k being almost n/r we have quadratic dependence on small parameters.…”

Section: Discussionmentioning

confidence: 99%

Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

Balogh

Cherkashin

Kiselev

2019

European Journal of Combinatorics

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We suggest a new method on coloring generalized Kneser graphs based on hypergraphs with high discrepancy and small number of edges. The main result is providing a proper coloring of K(n, n/2 − t, s) in (4 + o(1))(s + t) 2 colors, which is produced by Hadamard matrices. Also, we show that for colorings by independent set of a natural type, this result is the best possible up to a multiplicative constant.Our method extends to Kneser hypergraphs as well.

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Section: Discussionmentioning

confidence: 99%

Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

Balogh

Cherkashin

Kiselev

2019

European Journal of Combinatorics

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“…χ([G, r]) ≥ ω(G, r) r − 1 This contains as a special case the result of Alon, Frankl, and Lovász [4] on the chromatic number of Kneser hypergraphs, conjectured by Erdős [13]. A simpler proof of Theorem 4.3 using the Z/p-index is due to Matoušek [26]. Combinatorial proofs of this result and several of its variants and extensions are due to Ziegler [42].…”

Section: Generalizations To Hypergraph Coloringsmentioning

confidence: 99%

Intersection patterns of finite sets and of convex sets

Frick

2016

Proc. Amer. Math. Soc.

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Abstract. The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverbergtype theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant maps yields lower bounds for chromatic numbers. We obtain an essentially elementary proof of the result of Schrijver on the chromatic number of stable Kneser graphs. In fact, we show that every neighborly even-dimensional polytope yields a small induced subgraph of the Kneser graph of the same chromatic number.We furthermore use this geometric viewpoint to give tight lower bounds for the chromatic number of certain small subhypergraphs of Kneser hypergraphs.

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“…(For k = 2 this is equivalent to the Borsuk-Ulam theorem). In view of the purely combinatorial hypergraph coloring results proved with this tool (see Alon, Frankl & Lovász [1], Matoušek [10], Ziegler [21], etc. ), one is led to ask for an analogous combinatorial treatment of Dold's theorem, for a "Z k -Tucker lemma", etc.…”

Section: The Classical Case K =mentioning

confidence: 99%

Combinatorial Stokes formulas via minimal resolutions

Hanke

Sanyal

Schultz

et al. 2009

Journal of Combinatorial Theory, Series A

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We describe an explicit chain map from the standard resolution to the minimal resolution for the finite cyclic group Z_k of order k. We then demonstrate how such a chain map induces a "Z_k-combinatorial Stokes theorem", which in turn implies "Dold's theorem" that there is no equivariant map from an n-connected to an n-dimensional free Z_k-complex. Thus we build a combinatorial access road to problems in combinatorics and discrete geometry that have previously been treated with methods from equivariant topology. The special case k=2 for this is classical; it involves Tucker's (1949) combinatorial lemma which implies the Borsuk-Ulam theorem, its proof via chain complexes by Lefschetz (1949), the combinatorial Stokes formula of Fan (1967), and Meunier's work (2006).Comment: 18 page

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On the chromatic number of Kneser hypergraphs

Abstract: Abstract. We give a simple and elementary proof of Kříž's lower bound on the chromatic number of the Kneser r-hypergraph of a set system S.

Cited by 10 publications

References 12 publications

Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

Coloring general Kneser graphs and hypergraphs via high-discrepancy hypergraphs

Intersection patterns of finite sets and of convex sets

Combinatorial Stokes formulas via minimal resolutions

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