On the chromatic number of generalized Kneser hypergraphs

Abstract

The generalized Kneser hypergraph KG r (n, k, s) is the hypergraph whose vertices are all the k-subsets of {1, . . . , n}, and edges are r-tuples of distinct vertices such that any pair of them has at most s elements in their intersection. In this note, we show that for each non-negative integers k, n, r, s satisfying n ≥ r(k − 1) + 1, k > s ≥ 0, and r ≥ 2, we havewhich extends the previously known result by Alon-Frankl-Lovász.

On the number of star‐shaped classes in optimal colorings of Kneser graphs

“…About two decades later, Lovász [13] proved Kneser's conjecture by taking an extraordinary approach, using tools from algebraic topology. Since then, a lot of attention has been drawn to study various problems related to this conjecture, including a large number of new proofs, such as [5,10,12,14], and many generalizations, such as [2,4,6,7,9,11,15]. In particular, Aisenberg et al [1] gave a new proof of this conjecture using a simple counting argument based on the Hilton-Milner theorem, for all but finitely many cases.…”

Dold’s theorem from viewpoint of strong compatibility graphs

“…Several extensions of the octahedral Tucker lemma with fascinating applications in various area such as graph colorings, and fair division problems are known. We refer the interested reader to [10,13,28] for generalizations, and [1,9,10,20,22] for applications. Not long ago, a new generalization of octahedral Tucker's lemma, called G-Tucker's lemma, was introduced in [6].…”