Abstract colorings, games and ultrafilters

During the talk we consider various kinds of Ramsey-type theorems. Bergelson and Hindman investigated finite colorings of the complete graph [N]^2 with vertices in natural numbers, involving an algebraic structure of N. It follows from their result that for each finite coloring of [N]^2, there are finite pairwise disjoint sets F1, F2, … such that each set Fn contains an arithmetic progression of length n and all edges between vertices from different sets Fn have the same color. Colorings of graphs appear also in the context of combinatorial covering properties. Scheepers proved that a set of reals X is Menger if and only if for every finite coloring of the complete graph whose vertices are open sets in X and an open omega-cover U of X (i.e., every finite subset of X is contained in a proper subset of X from the cover), there are finite pairwise disjoint subfamilies F1, F2, … of U such that the union of these families is point-infinite cover of X and all edges between vertices from different sets Fn have the same color. The aim of the talk is to present a theorem that captures many results in a similar spirit (including mentioned above). To this end we use topological games and some special ultrafilters in the Stone—Cech compactification of semigroups. The research was motivated by the recent result of Tsaban, who extended the celebrated Hindman Finite Sum Theorem (and its high-dimensional version due to Milliken and Taylor) to covers of Menger spaces. The details of the Zoom meeting will be sent separately.