Borel chromatic number of closed graphs
We construct, for each countable ordinal $ξ$, a closed graph with Borel chromatic number two and Baire class $ξ$ chromatic number $\aleph\_0$.
math.LO↗
arXiv subjects
Dominique Lecomte
Publications and source records attributed to Dominique Lecomte.
Essential countability of treeable equivalence relations
We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.
math.LO↗
Descriptive complexity of countable unions of Borel rectangles
We give, for each countable ordinal $ξ\geq 1$, an example of a ${\bfΔ}^0_2$ countable union of Borel rectangles that cannot be decomposed into countably many ${\bfΠ}^0_ξ$ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ${\bfΔ}^0_ξ$-measurable countable coloring.
math.LO↗
Baire-class $ξ$ colorings: the first three levels
The $\mathbb{G}_0$-dichotomy due to Kechris, Solecki and Todor\vcević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb{G}_0$-dichotomy for $\boraxi$-measurable countable colorings when $ξ\leq 3$. A $\boraxi$-measurable countable coloring gives a covering of the diagonal consisting of countably many $\boraxi$ squares. This leads to the study of countable unions of $\boraxi$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $ξ\leq 2$.
math.LO↗
Baire class one colorings and a dichotomy for countable unions of $F_σ$ rectangles
We study the Baire class one countable colorings, i.e., the countable partitions into $F_σ$ sets. Such a partition gives a covering of the diagonal into countably many $F_σ$ squares. This leads to the study of countable unions of $F_σ$ rectangles. We give a Hurewicz-like dichotomy for such countable unions.
math.LO↗
Potential Wadge classes
Let $\bfΓ$ be a Borel class, or a Wadge class of Borel sets, and $2\leq d\leqω$ a cardinal. We study the Borel subsets of ${\mathbb R}^d$ that can be made $\bfΓ$ by refining the Polish topology on the real line. These sets are called potentially $\bfΓ$. We give a test to recognize potentially $\bfΓ$ sets.
math.LO↗
Decision Problems For Turing Machines
We answer two questions posed by Castro and Cucker, giving the exact complexities of two decision problems about cardinalities of omega-languages of Turing machines. Firstly, it is $D_2(Σ_1^1)$-complete to determine whether the omega-language of a given Turing machine is countably infinite, where $D_2(Σ_1^1)$ is the class of 2-differences of $Σ_1^1$-sets. Secondly, it is $Σ_1^1$-complete to determine whether the omega-language of a given Turing machine is uncountable.
cs.LO↗
Classical and Effective Descriptive Complexities of omega-Powers
We prove that, for each non null countable ordinal alpha, there exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete omega-powers, extending previous works on the topological complexity of omega-powers. We prove effective versions of these results. In particular, for each non null recursive ordinal alpha, there exists a recursive finitary language A such that A^omega is Sigma^0_alpha-complete (respectively, Pi^0_alpha-complete). To do this, we prove effective versions of a result by Kuratowski, describing a Borel set as the range of a closed subset of the Baire space by a continuous bijection. This leads us to prove closure properties for the classes Effective-Pi^0_alpha and Effective-Sigma^0_alpha of the hyperarithmetical hierarchy in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered by the second author in [Omega-Powers and Descriptive Set Theory, Journal of Symbolic Logic, Volume 70 (4), 2005, p. 1210-1232].
math.LO↗
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.
math.LO↗
How can we recognize potentially ${\bfΠ}^0_ξ$ subsets of the plane?
Let $ξ\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bfΠ}^0_ξ$ by refining the Polish topology on the real line. These sets are called potentially ${\bfΠ}^0_ξ$. We give a Hurewicz-like test to recognize potentially ${\bfΠ}^0_ξ$ sets.
math.LO↗
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.
cs.LO↗
Tests à la Hurewicz dans le plan
We give, for some Borel sets of a product of two Polish spaces, including the Borel sets with countable sections, a Hurewicz-like characterization of those which cannot become a transfinite difference of open sets by changing the two Polish topologies.
math.LO↗
Uniformisations partielles et critères à la Hurewicz dans le plan
We give characterizations of the Borel sets potentially in some Wadge class, among the Borel sets with countable vertical sections of a product of two Polish spaces. To do this, we use some partial uniformization results.
math.LO↗
Classes de Wadge potentielles des boréliens à coupes dénombrables
We give, for each non self-dual Wadge class C contained in the class of the Gdelta sets, a characterization of Borel sets which are not potentially in C, among Borel sets with countable vertical sections; to do this, we use results of partial uniformization.
math.LO↗
Classes de Wadge potentielles et théorèmes d'uniformisation partielle
We want to give a construction as simple as possible of a Borel subset of a product of two Polish spaces. This introduces the notion of potential Wadge class. Among other things, we study the non-potentially closed sets, by proving Hurewicz-like results. This leads to partial uniformization theorems, on big sets, in the sense of cardinality or Baire category.
math.LO↗
Complexité des boréliens à coupes dénombrables
We give, for each level of complexity L, a Hurewicz-like characterization of the Borel subsets with countable sections of a product of two Polish spaces that cannot become in L by changing the two Polish topologies.
math.LO↗
On minimal non-potentially closed subsets of the plane
We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todorcevic dichotomy about analytic graphs.
math.LO↗
Omega-powers and descriptive set theory
We study the sets of the infinite sentences constructible with a dictionary over a finite alphabet, from the viewpoint of descriptive set theory. Among other things, this gives some true co-analytic sets. The case where the dictionary is finite is studied and gives a natural example of a set at the level omega of the Wadge hierarchy.
math.LO↗