Search for: All records

Abstract We present an alternative proof that from large cardinals, we can force the tree property at $$\kappa ^+$$ and $$\kappa ^{++}$$ simultaneously for a singular strong limit cardinal $$\kappa $$ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $$\kappa =\aleph _{\omega ^2}$$ . 

Abstract We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals. 

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.