Baire class one colorings and a dichotomy for countable unions of FσF_\sigma rectangles

We study the Baire class one countable colorings, i.e., the countable partitions into $F_\sigma$ sets. Such a partition gives a covering of the diagonal into countably many $F_\sigma$ squares. This leads to the study of countable unions of $F_\sigma$ rectangles. We give a Hurewicz-like dichotomy for such countable unions

A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension

We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of Baire-measurable homomorphism

How can we recognize potentially Πξ0{\bf\Pi}^0_\xi subsets of the plane?

Let $\xi\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bf\Pi}^0_\xi$ by refining the Polish topology on the real line. These sets are called potentially ${\bf\Pi}^0_\xi$. We give a Hurewicz-like test to recognize potentially ${\bf\Pi}^0_\xi$ sets

Injective tests of low complexity in the plane

We study injective versions of the characterization of sets potentially in a Wadge class of Borel sets, for the first Borel and Lavrentieff classes. We also study the case of oriented graphs in terms of continuous homomorphisms, injective or not

Borel chromatic number of closed graphs

We construct, for each countable ordinal $\xi$, a closed graph with Borel chromatic number two and Baire class $\xi$ chromatic number $\aleph\_0$.Comment: The proof of the main lemma has been changed, and the main result is now better than in the previous versio

Topological Complexity of omega-Powers : Extended Abstract

This is an extended abstract presenting new results on the topological complexity of omega-powers (which are included in a paper "Classical and effective descriptive complexities of omega-powers" available from arXiv:0708.4176) and reflecting also some open questions which were discussed during the Dagstuhl seminar on "Topological and Game-Theoretic Aspects of Infinite Computations" 29.06.08 - 04.07.08