Andrzej Ruciński

A Dirac-Type Theorem for 3-Uniform Hypergraphs

A Hamiltonian cycle in a 3-uniform hypergraph is a cyclic ordering of the vertices in which every three consecutive vertices form an edge. In this paper we prove an approximate and asymptotic version of an analogue of Diracs celebrated theorem for graphs: for each γ>0 there exists n0 such that every 3-uniform hypergraph on

Threshold functions for Ramsey properties

n\geq n_0

An approximate Dirac-type theorem for k-uniform hypergraphs

vertices, in which each pair of vertices belongs to at least

When are small subgraphs of a random graph normally distributed

(1/2+\gamma)n

UPPER TAILS FOR SUBGRAPH COUNTS IN RANDOM GRAPHS

edges, contains a Hamiltonian cycle.

Random Graph Processes with Degree Restrictions

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 graph chosen uniformly from among all graphs with n vertices and N edges. For a fixed graph G and an integer r we address the question what is the minimum N = N (G, r, n) such that the random graph K(n, N) contains, almost surely, a monochromatic copy of G in every r-coloring of its edges ( K(n, N) -+ (G), , in short). We find a graph parameter )I = )Ie G) yielding { 0 if N < cn1 , llli.~ Prob(K(n, N) -+ (G),) = I if N > Cn1 , for some c, C > o. We use this to derive a number of consequences that deal with the existence of sparse Ramsey graphs. For example we show that for all r ~ 2 and k ~ 3 there exists C > 0 such that almost all graphs H with n vertices and cntfi edges which are Kk+l-free, satisfy H -+ (Kk),. We also apply our method to the problem of finding the smallest N = N(k, r, n) guaranteeing thatalmost all sequences I

Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels

a l < a2 < ... < aN

On graphs with linear Ramsey numbers

n contain an arithmetic progression of length k in every r-coloring, and show k-2 that N = 6( n k=T) is the threshold. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, EMORY UNIVERSITY, ATLANTA, GEORGIA 30322 E-mail address: rodlOmathcs. emory. edu E-mail address:rucinskiClvm.amu.edu.pl License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Ramsey properties of random graphs

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 all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges.

Perfect matchings in uniform hypergraphs with large minimum degree

SummaryLetG be a graph and letXn count copies ofG in a random graphK(n,p). The random variable