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For a ®xed graph H, the Ramsey number r (H) is de®ned to be the least integer N such that in any 2-coloring of the edges of the complete graph K N , some monochromatic copy of H is always formed. Let r(n, Á) denote the class of graphs H having n vertices and maximum degree at most Á. It was shown by Chvata  l, Ro È dl, Szemere  di, and Trotter that for(More)
We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most ∆ is less than 8(8∆) ∆ |V (H)|. This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erd˝ os. Applying the probabilistic method we also show that(More)
We establish a new lower bound on the l-wise collective minimum degree which guarantees the existence of a perfect matching in a k-uniform hypergraph, where 1 ≤ l < k/2. For l = 1, this improves a long standing bound by Daykin and Häggkvist [4]. Our proof is a modification of the approach of Han, Person, and Schacht from [8]. In addition, we fill a gap left(More)
A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian. We prove an approximate version of an analogous result for uniform hyper-graphs: For every k ≥ 3 and γ > 0, and for(More)
We study the following one-person game against a random graph: the Player's goal is to 2-colour a random sequence of edges e1, e2,. .. of a complete graph on n vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochro-matic triangle is created. The online version of the game requires that the Player should colour(More)
In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erd˝ os on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using(More)
A perfect matching in a k-uniform hypergraph on n vertices, n divisible by k, is a set of n/k disjoint edges. In this paper we give a sufficient condition for the existence of a perfect matching in terms of a variant of the minimum degree. We prove that for every k ≥ 3 and sufficiently large n, a perfect matching exists in every n-vertex k-uniform(More)