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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 Â l, Ro È dl, Szemere Â 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)

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)

Given a graph H and an integer r ≥ 2, let G → (H, r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = max F ⊆G |E(F)|/|V (F)| and define the Ramsey density m inf (H, r) as the infimum of m(G) over all graphs G such that G → (H, r). In the first part of this paper… (More)