Andrzej Rucinski

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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 KN, some monochromatic copy of H is always formed. Let H(n, ) denote the class of graphs H having n vertices and maximum degree at most . It was shown by Chvata l, RoÈdl, SzemereÂdi, and Trotter that for each there(More)
Probabilistic methods have been used to approach many problems<lb>of Ramsey theory. In this paper we study Ramsey type questions from the point<lb>of view of random structures.<lb>Let K(n, N) be the random graph chosen uniformly from among all graphs<lb>with n vertices and N edges. For a fixed graph G and an integer r we address<lb>the question what is the(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 Erdős 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)
The application of Stein’s method of obtaining rates of convergence to the normal distribution is illustrated in the context of random graph theory. Problems which exhibit a dissociated structure and problems which do not are considered. Results are obtained for the number of copies of a given graph G in K(n, p), for the number of induced copies of G, for(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 hypergraphs: For every k ≥ 3 and γ > 0, and for(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)
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 Erdős. Applying the probabilistic method we also show that for all(More)