An approximate and asymptotic version of an analogue of Dirac's celebrated theorem for graphs is proved: for each γ>0 there exists n0 such that every 3-uniform hypergraph on n_0 vertices, in which each pair of vertices belongs to at least $(1/2+\gamma)n$ edges, contains a Hamiltonian cycle.Expand

Lethr(n) denote the maximum number of edges in anr-uniform hypergraph onn vertices and in which the union of any three edges has size greater than 3r − 3.Expand

A negative answer is given by showing that 1 - \frac{1}{{l^{r - 1} }} is not a jump ifr≧3,l>2r, and edges, wherec=c(α) does not depend on ε andm.Expand

It is shown that for all graphs G, a function from the set of copies of in G to [0, 1] is a fractional -packing of G if for every edge e of G is defined to be the maximum value of over all fractional-packings of G.Expand

Probabilistic methods have been used to approach many problems of Ramsey theory. In this paper we study Ramsey type questions from the point of view of random structures. Let K(n, N) be the random… Expand

It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2), and it is proved that such graphs have necessarily low edge density.Expand