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

Combinatorial aspects of symmetries on groups

An MSc dissertation by Shivani Singh. University of Witwatersrand Faculty of Science, School of Mathematics. August 2016.These symmetries have interesting applications to enumerative combinatorics and to Ramsey theory. The aim of this thesis will be to present some important results in these fields. In particular, we shall enumerate the r-ary symmetric bracelets of length n.LG201

Thin subsets of groups

For a group $G$ and a natural number $m$, a subset $A$ of $G$ is called $m$-thin if, for each finite subset $F$ of $G$, there exists a finite subset $K$ of $G$ such that $|Fg\cap A|\leqslant m$ for every $g\in G\setminus K$. We show that each $m$-thin subset of a group $G$ of cardinality $\aleph_n$, $n= 0,1,...$ can be partitioned into $\leqslant m^{n+1}$ 1-thin subsets. On the other side, we construct a group $G$ of cardinality $\aleph_\omega$ and point out a 2-thin subset of $G$ which cannot be finitely partitioned into 1-thin subsets

A nilpotent IP polynomial multiple recurrence theorem

We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate

Ramsey functions for spaces with symmetries

In this dissertation we study the notion of symmetry on groups, topological spaces, et cetera. The relationship between such structures with symmetries and Ramsey Theory is re ected by certain natural functions. We give a general picture of asymptotic behaviour of these functions

Selective survey on Subset Combinatorics of Groups

We survey recent results concerning the combinatorial size of subsets of groups. For a cardinal k, according to its arrangement in a group G, a subset of G is distinguished as k-large, k-small, k-thin, k-thick and Pk-small. By analogy with topology, there arise the following combinatorial cardinal invariants of a group: density, cellularity, resolvability, spread etc. The paper consists of 7 sections: Ballean context, Amenability, Ideals, Partitions, Packings, Around thin subsets, Colorings