The Solecki submeasures and densities on groups

We introduce the Solecki submeasure $\sigma(A)=\inf_F\sup_{x,y\in G}|F\cap xAy|/|F|$ and its left and right modifications on a group $G$, and study the interplay between the Solecki submeasure and the Haar measure on compact topological groups. Also we show that the right Solecki density on a countable amenable group coincides with the upper Banach density $d^*$ which allows us to generalize some fundamental results of Bogoliuboff, Folner, Cotlar and Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson and Fish about the sumsets to the class of all amenable groups.Comment: 34 page

Randomness extraction and asymptotic Hamming distance

We obtain a non-implication result in the Medvedev degrees by studying sequences that are close to Martin-L\"of random in asymptotic Hamming distance. Our result is that the class of stochastically bi-immune sets is not Medvedev reducible to the class of sets having complex packing dimension 1

Graph Theory

[no abstract available

Axioms for infinite matroids

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.Comment: 33 pp., 2 fig

Locally finite graphs with ends: A topological approach, I. Basic theory

AbstractThis paper is the first of three parts of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form from the ArXiv [18]. They are complemented by a third part [28], which looks at the theory from an algebraic-topological point of view.The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the second part of this survey [19] will concentrate on those applications, this first part explores the new theory as such: it introduces the basic concepts and facts, describes some of the proof techniques that have emerged over the past 10 years (as well as some of the pitfalls these proofs have in stall for the naive explorer), and establishes connections to neighbouring fields such as algebraic topology and infinite matroids. Numerous open problems are suggested