Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...

Let S be a finite set of positive integers. A mixed hypergraph H is a onerealization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, Y. Chang and K. Wang, The smallest one-realization of a given set II, Discrete Math. 312 (2012) 2946–2951], we determined the minimum number of vertices of a 3-uniform bi-hypergraph which is a one-real...

A partial Steiner (k, l)-system is a k-uniform hypergraph G with the property that every l-element subset of V is contained in at most one edge of G . In this paper we show that for given k, l and t there exists a partial Steiner (k, l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of t...

For an r-uniform hypergraph G define N(G, l; 2) (N(G, l;Zn)) as the smallest integer for which there exists an r-uniform hypergraph H on N(G, l; 2) (N(G, l;Zn)) vertices with clique(H)< l such that every 2-coloring (Zn-coloring) of the edges of H implies a monochromatic (zero-sum) copy of G. Our results strengthen a Ramsey-type theorem of Bialostocki and Dierker on zero-sum hypermatchings. As a...

We study thresholds for the appearance of a 2-core in random hypergraphs that are a mixture of a constant number of random uniform hypergraphs each with a linear number of edges but with different edge sizes. For the case of two overlapping hypergraphs we give a solution for the optimal (expected) number of edges of each size such that the 2-core threshold for the resulting mixed hypergraph is ...

Let F be an r-uniform hypergraph with f vertices, where f > r ≥ 3. In [12], R. Yuster posed the problem of whether there exists an algorithm which, for a given r-uniform hypergraph H with n vertices, computes the number of induced copies of F in H in time o(n ). The analogous question for graphs (r = 2) was known to hold from a O(nf−ε) time algorithm of Nešetřil and Poljak [9] (for a constant ε...

Let C (3) n denote the 3-uniform tight cycle, that is the hypergraph with vertices v1, . . . , vn and edges v1v2v3, v2v3v4, . . . , vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red-blue coloring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n...

A multi-bridge hypergraph is an h-uniform linear hypergraph consisting of some linear paths having common extremities. In this paper it is proved that the multisets of path lengths of two chromatically equivalent multi-bridge hypergraphs are equal provided the multiplicities of path lengths are bounded above by 2h−1 − 2. ∗This research is partially supported by Abdus Salam School of Mathematica...

For a pair of integers 1Fgr, the g-chromatic number of an r-uniform Ž . hypergraph Hs V, E is the minimal k, for which there exists a partition of V into subsets < < T , . . . , T such that elT Fg for every egE. In this paper we determine the asymptotic 1 k i Ž . behavior of the g-chromatic number of the random r-uniform hypergraph H n, p for all r Ž yrq1. possible values of g and for all value...

In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a k-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval [0, 2], and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H+-eigenvalues of the Laplacian and all the smallest H+-eigenvalues of its sub-t...