Let 2 ≤ q ≤ min{p, t − 1} be fixed and n → ∞. Suppose that F is a p-uniform hypergraph on n vertices that contains no complete q-uniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of F in many cases. For example, when q = 2 and p ∈ {t, t+ 1}, the maximum is ( n t−1 ) t−1 + o(nt−1), and when p = t = 3, it is b (n−1) 2 4 c for all n ≥ 3. Our proofs use the Krusk...

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible para...

A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k. As a consequence of our result, we extend bound...

A graph is one-regular if its automorphism group acts regularly on the arc set. In this paper, we construct a new infinite family of one-regular Cayley graphs of any prescribed valency. In fact, for any two positive integers , k 2 except for ( , k) ∈ {(2,3), (2,4)}, the Cayley graph Cay(Dn,S) on dihedral groups Dn = 〈a, b | an = b2 = (ab)2 = 1〉 with S = {a1+ +···+ t b | 0 t k − 1} and n = ∑k−1 ...

An [n, k, r]-hypergraph is a hypergraph :Yf = (V, E) whose vertex set V is partitioned into n k-element sets V1, V2, ... , Vn and for which, for each choice of r indices, 1 :::;; i1 < i2 < ... < ir :::;; n, there is exactly one edge e E E such that len Vii = 1 if i E {i1, i2, ... , ir} and otherwise le n Vii = 0. An independent transversal of an [n, k, r ]-hypergraph is a set T = {a1,a2, .. . ,...

Let S be a set of transpositions generating the symmetric group Sn, where n ≥ 3. It is shown that if the girth of the transposition graph of S is at least 5, then the automorphism group of the Cayley graph Cay(Sn, S) is the direct product Sn×Aut(T (S)), where T (S) is the transposition graph of S; the direct factors are the right regular representation of Sn and the image of the left regular ac...

The purpose of this paper is the study of Cayley graph associated to a semihypergroup(or hypergroup). In this regards first  we associate a Cayley graph to every semihypergroup and then we study theproperties of this graph, such as  Hamiltonian cycles in this  graph.  Also, by some of examples we will illustrate  the properties and behavior of  these Cayley  graphs, in particulars we show that ...

In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let K (k) t be the complete kuniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(n) copies of ...