14,384 research outputs found

### Baire class one colorings and a dichotomy for countable unions of $F_\sigma$ 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

### There Exist some Omega-Powers of Any Borel Rank

Omega-powers of finitary languages are languages of infinite words
(omega-languages) in the form V^omega, where V is a finitary language over a
finite alphabet X. They appear very naturally in the characterizaton of regular
or context-free omega-languages. Since the set of infinite words over a finite
alphabet X can be equipped with the usual Cantor topology, the question of the
topological complexity of omega-powers of finitary languages naturally arises
and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). It
has been recently proved that for each integer n > 0, there exist some
omega-powers of context free languages which are Pi^0_n-complete Borel sets,
that there exists a context free language L such that L^omega is analytic but
not Borel, and that there exists a finitary language V such that V^omega is a
Borel set of infinite rank. But it was still unknown which could be the
possible infinite Borel ranks of omega-powers. We fill this gap here, proving
the following very surprising result which shows that omega-powers exhibit a
great topological complexity: for each non-null countable ordinal alpha, there
exist some Sigma^0_alpha-complete omega-powers, and some Pi^0_alpha-complete
omega-powers.Comment: To appear in the Proceedings of the 16th EACSL Annual Conference on
Computer Science and Logic, CSL 2007, Lausanne, Switzerland, September 11-15,
2007, Lecture Notes in Computer Science, (c) Springer, 200

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