On Galvin's lemma and Ramsey spaces

An abstract version of Galvin's lemma is proven, within the framework of the theory of Ramsey spaces. Some instances of it are explored.Comment: Accepted in Annals of Combinatoric

Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure

Asymptotic dimension and small subsets in locally compact topological groups

We prove that for a coarse space $X$ the ideal $S(X)$ of small subsets of $X$ coincides with the ideal $D_<(X)$ of subsets $A\subset X$ of asymptotic dimension $asdim(A)<asdim(X)$ provided that $X$ is coarsely equivalent to an Euclidean space $R^n$. Also we prove that for a locally compact Abelian group $X$, the equality $S(X)=D_<(X)$ holds if and only if the group $X$ is compactly generated.Comment: 10 page

Semialgebraic Graphs having Countable List-Chromatic Numbers

The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.Comment: This version has been completely rewritten. It will appear in PAM