A generalized kneser conjecture

Generalized Kneser coloring theorems with combinatorial proofs

“…An alternative proof was later given by Bárány [Bár78], extensions by Schrijver [Sch78], Alon, Frankl & Lovász [AFL86], Dol'nikov [Dol'88], Sarkaria [Sar90], and finally by Kriz [Kri92,Kri00], whose result implies the theorems by Lovász, Dol'nikov and Alon-Frankl-Lovász. All of these were proved using Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions.…”

“…This theorem, in combination with Lemma 3.2, has many well-known special cases, for constant s: Kri00], and S = [n] k : Sarkaria [Sar90]. The generalization to non-constant s is not done for it's own interest, but since it is needed for the first part of our proof, where we show that one may assume that n − 1 is divisible by r − 1.…”

“…• a simple combinatorial proof of Dol'nikov's theorem (an extension of Matoušek's proof [Mat01] of the Kneser conjecture), • a new, fairly general hypergraph coloring theorem (which has the theorems by Lovász [Lov78], Alon, Frankl & Lovász [AFL86], Dol'nikov [Dol'88], Sarkaria [Sar90], and Kriz [Kri92,Kri00] as special cases), together with a combinatorial proof, and • a combinatorial proof of Schrijver's theorem (via cyclic oriented matroids).…”

On the chromatic number of Kneser hypergraphs

“…This result was further extended by Ziegler [15], who proved a common generalization of Theorem 1.1 and of theorems of Alon et al [1] and of Sarkaria [14]. The proof is based on topological ideas but uses no "continuous" structure.…”

“…This was proved in 1978 by Lovász [12], as one of the earliest and most spectacular applications of topological methods in combinatorics. Several other proofs have been published since then, all of them topological; among them, at least those of Bárány [2], Dol'nikov [6] (also see [5] and [7]), and Sarkaria [14] can be regarded as substantially different from each other and from Lovász' original proof. Erdős' generalization of Kneser's conjecture to hypergraphs, dealing with the chromatic number of KG r (…”