Generalized Kneser coloring theorems with combinatorial proofs

Details Generalized Kneser coloring theorems with combinatorial proofs Generalized Kneser coloring theorems with combinatorial proofs

2001-03-23

arxiv.org/abs/math/0103146v1

Mathematics

Combinatorics

19 pages, 4 figures

Scientific paper

10.1007/s002220100188

The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.

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