Colorful Subhypergraphs in Uniform Hypergraphs

Abstract

There are several topological results ensuring in any properly colored graph the existence of a colorful complete bipartite subgraph, whose order is bounded from below by some topological invariants of some topological spaces associated to the graph. Meunier [Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014] presented the first colorful type result for uniform hypergraphs. In this paper, we give some new generalizations of the $\mathbb{Z}_p$-Tucker lemma and by use of them, we improve Meunier's result and some other colorful results by Simonyi, Tardif, and Zsbán [Colourful theorems and indices of homomorphism complexes, The Electronic Journal of Combinatorics, 2014] and by Simonyi and Tardos [Colorful subgraphs in Kneser-like graphs, European Journal of Combinatorics, 2007] to uniform hypergraphs. Also, we introduce some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs. A hierarchy between these lower bounds is presented as well.

A new lower bound for the chromatic number of general Kneser hypergraphs

“…Some extensions of this result to more general topological settings can be found in [2,18]. These results as well as Meunier's were extended to the case of uniform hypergraphs in [1].…”

Multilabeled Versions of Sperner's and Fan's Lemmas and Applications

“…In the next section, we will provide yet another proof of this result, with an approach that clearly departs from the other ones. A kind of "dual" statement of Item (1) is present in Section 6 of the cited paper by Frick et al However, Item (2) seems to be the natural "dual" version, and we get it with almost the same proof as for Item (1).…”

“…This case is even simpler. Choose the preferences such that every player has an indifference point x in [0, 1]: a point for which the player is indifferent between choosing [0, x] and [x, 1], and such that if the cake is cut at any point y = x, then that player must prefer the piece that contains x. Consider 4 players with distinct indifference points:…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…The resulting Z q -Fan lemmas actually provide combinatorial proofs of Dold's theorem. The deduction of the Z q -Fan lemmas from their Z q -Tucker versions works as explained above in the Z 2 case [15].…”