Ramsey theorem for classes of hypergraphs with forbidden complete subhypergraphs
Ramsey theoretic consequences of some new results about algebra in the Stone-Čech compactification.
Ramsey theory applications.
Ramsey theory for structures: Nešetřil's result on finite metric spaces.
Ramseyan properties of graphs.
Ramseyan ultrafilters
We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we...
Ramsey-like properties for bi-Lipschitz mappings of finite metric spaces
Let , be metric spaces and an injective mapping. We put ; , , and (the distortion of the mapping ). Some Ramsey-type questions for mappings of finite metric spaces with bounded distortion are studied; e.g., the following theorem is proved: Let be a finite metric space, and let , be given numbers. Then there exists a finite metric space , such that for every mapping ( arbitrary metric space) with one can find a mapping , such that both the mappings and have distortion at...
Ramsey-type theorems
Random Cayley graphs are expanders: a simple proof of the Alon-Roichman theorem.
Random directed trees and forest -- drainage networks with dependence.
Random even graphs.
Random Liouville functions and normal sets
Random mappings, forests, and subsets associated with Abel-Cayley-Hurwitz multinomial expansions.
Random matrices and permutations, matrix integrals and integrable systems
Random matrices, magic squares and matching polynomials.
Random networks with sublinear preferential attachment: degree evolutions.
Random orderings and unique ergodicity of automorphism groups
We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations...
Random orders and gambler's ruin.
Random Packings and Coverings in Random Trees
Random procedures for dominating sets in bipartite graphs
Using multilinear functions and random procedures, new upper bounds on the domination number of a bipartite graph in terms of the cardinalities and the minimum degrees of the two colour classes are established.