A coloring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose color does not repeat in e. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χCF (H). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph H with m edges, χCF (H) is at most of the order of rm lo...

We say a s-uniform r-partite hypergraph is complete, if it has a vertex partition {V1, V2, . . . , Vr} of r classes and its hyperedge set consists of all the s-subsets of its vertex set which have at most one vertex in each vertex class. We denote the complete s-uniform r-partite hypergraph with k vertices in each vertex class by Ts,r(k). In this paper we prove that if h, r and s are positive i...

Let m∗(n) be the minimum number of edges in an n-uniform simple hypergraph that is not two colorable. We prove that m∗(n) = Ω(4n/ ln(n)). Our result generalizes to r-coloring of b-simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b-simple n-uniform hypergraph that is not r-colorable must be Ω(rn/ ln(n)). By trimming arguments it implies that every such graph...

The extension of conventional clustering to hypergraph clustering, which involves higher order relations among nodes instead of pairwise relations, is increasingly gaining attention, as multi-node relations may capture more informative clustering structures. One of widely used methodologies for hypergraph clustering is to minimize a normalized sum of the costs to partition hyperedges across clu...

The r-uniform linear k-cycle C k is the r-uniform hypergraph on k(r−1) vertices whose edges are sets of r consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to sho...

The Harary-Sachs theorem for k-uniform hypergraphs equates the codegree-d coefficient of adjacency characteristic polynomial a uniform hypergraph with weighted sum subgraph counts over certain multi-hypergraphs d edges. We begin by showing that classical graphs is indeed special case this general theorem. To end we apply generalized to leading coefficients various hypergraphs. In particular, pr...

Edge colorings of r-uniform hypergraphs naturally define a multicoloring on the 2-shadow, i.e. on the pairs that are covered by hyperedges. We show that in any (r− 1)-coloring of the edges of an r-uniform hypergraph with n vertices and at least (1−2)(n r ) edges, the 2-shadow has a monochromatic matching covering all but at most o(n) vertices. This result implies that for any fixed r and suffic...

For a set of integers S, define ( S APk ) to be the k-uniform hypergraph with vertex set S and hyperedges corresponding to the set of all arithmetic progression of length k in S. Similarly, for a graph H, define ( H Kk ) to be the ( k 2 ) -uniform hypergraph on the vertex set E(H) with hyperedges corresponding to the edge sets of all copies of Kk in H. Also, we say that a k-uniform hypergraph h...

The theory of mixed hypergraph coloring was first introduced by Voloshin in 1993 and has been growing ever since. The proper coloring of a mixed hypergraph H = (X, C,D) is the coloring of the vertex set X so that no D-hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D, C, or B-hypergraph respectively wher...

An integral sequence d = (d1, d2, . . . , dn) is hypergraphic if there is a simple hypergraph H with degree sequence d, and such a hypergraph H is a realization of d. A sequence d is r-uniform hypergraphic if there is a simple r-uniform hypergraph with degree sequence d. Similarly, a sequence d is r-uniformmulti-hypergraphic if there is an r-uniformhypergraph (possibly with multiple edges) with...