arXiv subjects
Publications and source records attributed to Dominique Lecomte.
We introduce a family of forcing notions that are helpful in showing that certain graphs do not have countable colourings of (additive) Borel class alpha. We construct graphs that are ''weakly minimal'' for such colourings.
We study several natural classes of graphs on a zero-dimensional metrizable compact space having no continuous coloring. We compare these graphs with the quasi-order associated with injective continuous homomorphisms. We prove the existence of an antichain basis for these classes. We determine the size of such an antichain basis. We provide a concrete antichain basis when there is a countable one. We also provide some related quasi-orders and equivalence relations which are analytic complete as sets.
We provide, for each natural number $n$ and each class among $D_n(\Sigma^0_1)$, $\bar D_n(\Sigma^0_1)$ and $D_{2n+1}(\Sigma^0_1)\oplus\bar D_{2n+1}(\Sigma^0_1)$, a regular language whose associated omega-power is complete for this class.
The type $\tau$($\alpha$) of an irrational number $\alpha$ measures the extent to which rational numbers can closely approximate $\alpha$. More precisely, $\tau$($\alpha$) is the infimum over those t$\in$R for which |$\alpha$--h/k| 0. In this paper, we regard the type as a function $\tau$:R\Q$\rightarrow$[1,$\infty$] and explore its descriptive properties. We show that $\tau$ is invariant under the natural action of GL2(Q) on R\Q. We show that $\tau$ is densely onto, and we compute the descriptive complexity of the pre-image of the singletons and of certain intervals. Finally, we show that the function $\tau$ is [1,$\infty$]-upper semi-Baire class 1 complete.
We first consider the class K of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for K made up of countable graphs, comparing them with the quasi-order associated with injective continuous homomorphisms. We prove that the size of such a basis is sharp, using odometers. However, using odometers again, we prove that there is no antichain basis in K, and provide infinite descending chains in K. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of the Cantor space is Borel reducible to the equivalence relation associated with our quasi-order. We also prove that there is no antichain basis in the class of graphs on a zero-dimensional Polish space with continuous chromatic number at least three. We study the graphs induced by a continuous function, and show that any basis for the class of graphs induced by a homeomorphism of a zero-dimensional metrizable compact space with continuous chromatic number at least three must have size continuum, using odometers or subshifts.
The $\omega$-power of a finitary language L over a finite alphabet $\Sigma$ is the language of infinite words over $\Sigma$ defined by L $\infty$ := {w 0 w 1. .. $\in$ $\Sigma$ $\omega$ | $\forall$i $\in$ $\omega$ w i $\in$ L}. The $\omega$-powers appear very naturally in Theoretical Computer Science in the characterization of several classes of languages of infinite words accepted by various kinds of automata, like B{\"u}chi automata or B{\"u}chi pushdown automata. We survey some recent results about the links relating Descriptive Set Theory and $\omega$-powers.
We prove that, for any natural number n $\ge$ 1, we can find a finite alphabet $Σ$ and a finitary language L over $Σ$ accepted by a one-counter automaton, such that the $ω$-power L $\infty$ := {w 0 w 1. .. $\in$ $Σ$ $ω$ | $\forall$i $\in$ $ω$ w i $\in$ L} is $Π$ 0 n-complete. We prove a similar result for the class $Σ$ 0 n .
We study the class of analytic binary relations on Polish spaces, compared with the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is $Σ$ 0 $ξ$ (or $Π$ 0 $ξ$), when $ξ$ $\ge$ 3, by providing a concrete finite antichain basis. We give a similar characterization for arbitrary relations when $ξ$ = 1. When $ξ$ = 2, we provide a concrete antichain of size continuum made of locally countable Borel relations minimal among non-$Σ$ 0 2 (or non-$Π$ 0 2) relations. The proof of this last result allows us to strengthen a result due to Baumgartner in topological Ramsey theory on the space of rational numbers. We prove that positive results hold when $ξ$ = 2 in the acyclic case. We give a general positive result in the non-necessarily locally countable case, with another suitable acyclicity assumption. We provide a concrete finite antichain basis for the class of uncountable analytic relations. Finally, we deduce from our positive results some antichain basis for graphs, of small cardinality (most of the time 1 or 2).
We study the analytic digraphs of uncountable Borel chromatic number on Polish spaces, and compare them with the notion of injective Borel homomorphism. We provide some minimal digraphs incomparable with G 0. We also prove the existence of antichains of size continuum, and that there is no finite basis. 2010 Mathematics Subject Classification. 03E15, 54H05
We study the class of Borel equivalence relations under continuous reducibility. In particular , we characterize when a Borel equivalence relation with countable equivalence classes is $Σ$ 0 $ξ$ (or $Π$ 0 $ξ$). We characterize when all the equivalence classes of such a relation are $Σ$ 0 $ξ$ (or $Π$ 0 $ξ$). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G $δ$ .
We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.
We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${\bfΣ}^0_1 \!\times\! {\bfΣ}^0_ξ$ sets, or by a ${\bfΠ}^0_1 \!\times\! {\bfΠ}^0_ξ$ set.
The literature provides dichotomies involving homomorphisms (like the G 0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of analytic equivalence relations. This is not possible in general. We show that, under some acyclicity (and also topological) assumptions, this is widely possible. In particular, we prove that, for any non-self dual Borel class Γ, there is a concrete finite =< c-antichain basis for the class of Borel relations, whose closure has acyclic symmetrization, and which are not potentially in Γ. Along similar lines, we provide a sufficient condition for =< c-reducing G 0. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.
The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\_{δσ}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is ${\bfΠ}^1\_1$-complete, in a uniform way. We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes ${\bfΠ}^1\_1$ and ${\bfΣ}^1\_2$ in the theory of martingales.
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.
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $\Game (D\_n({\bfΣ}^0\_2))$ for some natural number $n\geq 1$, where $\Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^ω$ into the class ${\bfΔ}^1\_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${\bfΔ}^1\_2$.
We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $\mathbb{E}\_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $\cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $E\in {\cal F}$ or $F$ is Borel reducible to $E$, then $\cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
We construct, for each countable ordinal $ξ$, a closed graph with Borel chromatic number two and Baire class $ξ$ chromatic number $\aleph\_0$.