In a break-through paper, Lovász [20] determined the chromatic number of Kneser graphs. This was improved by Schrijver [27], by introducing the Schrijver subgraphs of Kneser graphs and showing that their chromatic number is the same as that of Kneser graphs. Alon, Frankl, and Lovász [2] extended Lovász's result to the usual Kneser hypergraphs and one of our main results is to extend this to a new family of general Kneser hypergraphs. Moreover, as a special case, we settle a question from Naserasr and Tardif [26]. In 2011, Meunier introduced almost 2-stable Kneser hypergraphs and determined their chromatic number as an approach to a supposition of Ziegler [35] and a conjecture of Alon, Drewnowski, and Łuczak [3]. In this work, our second main result is to improve this by computing the chromatic number of a large family of Schrijver hypergraphs. Our last main result is to prove the existence of a completely multicolored complete bipartite graph in every coloring of a graph which extends a result of Simonyi and Tardos [29]. JID:YJCTB AID:2930 /FLA [m1L; v1.156; Prn:10/06/2015; 9:53] P.2 (1-24) 2 M. Alishahi, H. Hajiabolhassan / J. Combin. Theory Ser. B ••• (••••) •••-••• The first two results are proved using a new improvement of the Dol'nikov-Kříž [7,18] bound on the chromatic number of general Kneser hypergraphs.

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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.

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