Jindřich Zapletal

Forcing with quotients

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

Four and more

We isolate several large classes of definable proper forcings and show how they include many partial orderings used in practice.

Ramsey theorems for product of finite sets with submeasures

We prove parametrized partition theorem on products of finite sets equipped with submeasures, improving the results of Di Prisco, Llopis, and Todorcevic.

Cofinalities of Borel ideals

We study the possible values of the cofinality invariant for various Borel ideals on the natural numbers. We introduce the notions of a fragmented and gradually fragmented ideal and prove a dichotomy for fragmented ideals. We show that every gradually fragmented ideal has cofinality consistently strictly smaller than the cardinal invariant and produce a model where there are uncountably many pairwise distinct cofinalities of gradually fragmented ideals.

Terminal notions in set theory

Abstract In mathematical practice certain formulas φ(x) are believed to essentially decide all other natural properties of the object x . The purpose of this paper is to exactly quantify such a belief for four formulas φ(x) , namely “ x is a Ramsey ultrafilter”, “ x is a free Souslin tree”, “ x is an extendible strong Lusin set” and “ x is a good diamond sequence”.

Why Y-c.c.

Abstract We outline a portfolio of novel iterable properties of c.c.c. and proper forcing notions and study its most important instantiations, Y-c.c. and Y-properness. These properties have interesting consequences for partition-type forcings and anticliques in open graphs. Using Neemans side condition method it is possible to obtain PFA variations and prove consistency results for them.

SEPARATION PROBLEMS AND FORCING

Certain separation problems in descriptive set theory correspond to a forcing preservation property, with a fusion type infinite game associated to it. As an application, it is consistent with the axioms of set theory that the circle 𝕋 can be covered by ℵ1 many closed sets of uniqueness while a much larger number of H-sets is necessary to cover it.