In this paper we define the concept of clique number of uniform hypergraph and study its relationship with circular chromatic number and clique number. For every positive integer k,p and q, 2q ≤ p we construct a k-uniform hypergraph H with small clique number whose circular chromatic number equals p q . We define the concept and study the properties of c-perfect k-uniform hypergraphs .

We prove that any k-uniform hypergraph on n vertices with minimum degree at least n 2(k−1) + o(n) contains a loose Hamilton cycle. The proof strategy is similar to that used by Kühn and Osthus for the 3-uniform case. Though some additional difficulties arise in the k-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.

We show that there is n0 such that if H is a 3-uniform hypergraph on n ∈ 4Z, n ≥ n0, vertices such that δ1(H) ≥ ( n−1 2 ) − ( 3n/4 2 ) + 3n 8 + 1 2 , then H can be tiled with copies of C4, the unique 3-uniform hypergraph on four vertices with two edges. The degree condition is tight when 8|n.

We investigate the threshold probability for the property that every r-coloring of the edges of a random binomial k-uniform hypergraph G(k)(n, p) yields a monochromatic copy of some fixed hypergraph G. In this paper we solve the problem for arbitrary k ≥ 3 and k-partite, k-uniform hypergraphs G.

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose thatH is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than ( n−1 2 )

Given a 3-uniform hypergraph F , let ex3(n,F) denote the maximum possible size of a 3-uniform hypergraph of order n that does not contain any subhypergraph isomorphic to F . Our terminology follows that of [16] and [10], which are comprehensive survey articles of Turán-type extremal graph and hypergraph problems, respectively. Also see the monograph of Bollobás [2]. There is an extensive litera...

The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. While independent sets in graphs are defined as sets of vertices containing no edges, hypergraphs have different types of independent sets depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is jindependent if its i...

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers ` ≥ k ≥ 2 and every d > 0 there exists % > 0 for which the following holds: if H is a sufficiently large kuniform hypergra...

The main purpose of this paper is to prove that if H is a 4-uniform hypergraph with n vertices and m edges, then the transversal number r(H) <2(m +n)/9. All standard terminology of hypergraphs is from [ 11. Suppose H = (V, E) is a k-uniform hypergraph with n vertices and m edges. Tuza [2] proposed the problem of finding an upper bound for the transversal number r(H), of the form t(H) < ck(n + m...

We show that if we color the hyperedges of the complete 3-uniform hypergraph on 2n + √ 18n + 1 + 2 vertices with n colors, then one of the color classes contains a loose path of length three. Let P denote the 3-uniform path of length three by which we mean the only connected 3-uniform hypergraph on seven vertices with the degree sequence (2, 2, 1, 1, 1, 1, 1). By R(P ;n) we denote the multicolo...