Generalized Kneser Coloring Theorems with Combinatorial Proofs

The Kneser conjecture (1955) was proved by Lovász (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, Matoušek 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-Lovász, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver’s theorem.