MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$ -divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$ -sphere. This answers a question by Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_{2}(\mathbb{Z})$ on $\mathsf{P}^{1}(\mathbb{R})$ . The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.

INFINITE COMBINATORICS PLAIN AND SIMPLE

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

A constructive way to compute the Tarski number of a group

The Tarski number of a group [Formula: see text] is the minimal number of the pieces of paradoxical decompositions of that group. Using configurations along with a matrix combinatorial property, we construct paradoxical decompositions. We also compute an upper bound for the Tarski number of a given non-amenable group by counting the number of paths in a diagram associated to the group.

ON CONFIGURATION GRAPH AND PARADOXICAL DECOMPOSITION

In this paper, we introduce the concept of configuration graph and show how one can use this notion to simplify the theorem proved by Rejali and Yousofzadeh [Configuration of groups and paradoxical decompositions, Bull. Belg. Math. Soc. Simon Stevin18 (2011) 157–172].

A generalization of Sierpiński's paradoxical decompositions: Coloring semialgebraic grids

A structure is an n-grid if each Ei, is an equivalence relation on A and whenever X and Y are equivalence classes of, respectively, distinct Ei, and Ej, then X ∩ Y is finite. A coloring χ: A → n is acceptable if whenever X is an equivalence class of Ei, then {x ∈ X: χ(x) = i} is finite. If B is any set, then the n-cube Bn = (Bn; E0, …, En−1) is considered as an n-grid, where the equivalence classes of Ei are the lines parallel to the i-th coordinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpihski [17], proved that ℝn has an acceptable coloring iff 2ℵ0 ≤ ℵn−2. The main result is: if is a semialgebraic (i.e., first-order definable in the field of reals) n-grid, then the following are equivalent: (1) if embeds all finite n-cubes, then 2ℵ0 ≤ ℵn−2: (2) if embeds ℝn, then 2ℵ0 ≤ ℵn−2; (3) has an acceptable coloring.