## Special Sesions

Set Theory

Invited Speakers

• Dominique Lecomte, Université de Picardie
Continuous 2-colorings and discrete dynamical systems

We consider various classes of graphs, from the most general ones to those induced by a function. The basic concern in this work is to understand when a graph has a continuous coloring in two colors. We compare the graphs with the quasi-order associated to either injective continuous homomorphisms, or continuous homomorphisms. We present structural properties of these quasi-orders. We will see that discrete dynamical systems are very useful to do that. This analysis also provides information about the quasi-order of Borel reducibility on the class of analytic equivalence relations, in particular about the relation of conjugacy of minimal homeomorphisms of the Cantor space.

• Victor Torres-Perez, Vienna University of Technology
The world without Martin's Axiom

The first of Hilbert's famous list of problems at the beginning of the 20th century was to establish Cantor's Continuum Hypothesis (CH), i.e. if there is no uncountable subset of the reals with cardinality strictly less than the continuum. After the works of Gödel and Cohen, it was concluded that the traditional axioms of Set Theory (ZFC) cannot decide CH.

Since then, new axioms have emerged. Prominently we have Forcing Axioms. One of the first Forcing Axioms ever considered was Martin's Axiom (MA). While MA implies the negation of the CH, it does not decide the exact value of the continuum. However, generalizations of MA like the Proper Forcing Axiom (PFA) or Martin's Maximum (MM) do imply that the continuum is the second uncountable cardinal. Besides, PFA or MM imply the negation of certain square principles or tree properties (among a very large number of interesting consequences). This means in particular that these axioms require the existence of large cardinals.

There are other relatively new principles, which have strong consequences similar to the ones from PFA or MM, but they can coexist consistently with the absence of MA or even imply this absence. A couple of these principles are, for example, Rado's Conjecture (RC) and the P-Ideal Dichotomy (PID).

We will give a general review of results involving these kinds of principles, including some of ours obtained along the previous years.

There, it is possible to observe that even if they can avoid MA, they are still quite powerful like these traditional Forcing Axioms. We will expose one of our last results, where we prove (with L. Wu) that PID implies the negation of a certain type of two-cardinals square principle.

• Diana Carolina Motoya, Universidad de Viena
Maximal almost disjoint families and singulars

Throughout the last years, many generalizations from classical cardinal characteristics of the Baire space have been studied. Particularly, special interest has been given to the study of the combinatorics of the generalized Baire spaces $\kappa^\kappa$ when $\kappa$ is an uncountable regular cardinal (or even a large cardinal). In this talk, I will present some results regarding a generalization to the context of singular cardinals of the concept of maximal almost disjointness. The first known result in this area is due to Erdös and Hechler in \cite{ErHe:Sad}, who introduced the concept of almost disjointness for families of subsets of a singular cardinal $\lambda$ and proved many interesting results: for instance, if $\lambda$ is a singular cardinal of cofinality $\kappa< \lambda$ and there is an almost disjoint family at $\kappa$ of size $\gamma$, then there is a maximal almost disjoint family at $\lambda$ of the same size.  The main result of this talk is the construction of a generic extension in which the inequality $\fra(\lambda)< \fra$ holds for $\lambda$ a singular cardinal of countable cofinality. The model combines the classical technique of Brendle to get a model in which $\frb < \fra$ together with the use of P\v{r}\'{i}kr\'{y} type forcings which change the cofinality of a given large cardinal $\kappa$ to be countable and, at the same time control the size of the power set of this given cardinal.

• Todor Tsankov, Université Claude Bernard – Lyon 1
Maximal highly proximal flows of locally compact groups

The notion of a highly proximal extension of a flow generalizes the one of an almost one-to-one extension (injective on a dense G_delta set), which is an important tool in topological dynamics. The existence of maximal such extensions was proved by Auslander and Glasner in the 70s for minimal flows using an abstract argument, and a concrete construction using near-ultrafilters was recently given by Zucker for arbitrary flows. When the acting group is discrete, the MHP extension is nothing but the Stone space of the Boolean algebra of the regular open sets of the space. We give yet another construction of the MHP extension for arbitrary topological groups and prove that for MHP flows of a locally compact group G, the stabilizer map x → G_x is continuous (for general flows, this map is only semi-continuous). This is a common generalization of a theorem of Frolík that the set of fixed points of a homeomorphism of a compact, extremally disconnected space is open and a theorem of Veech that the action of a locally compact group on its greatest ambit is free. This is joint work with Adrien Le Boudec.