Fraenkel–Mostowski models revisited. This paper isolates several properties of dynamical ideals and shows how they are reflected in the associated permutation models of ZF.

Classifying open subgroups of non-archimedean groups   a completely preliminary version.

Triangles and Vitali sets It is consistent with ZF+DC that the hypergraph of equilateral triangles in the plane has countable chromatic number while there is no Vitali set.

 Coloring equilateral triangles. I show it consistent with ZF+DC that the hypergraph of equilateral triangles in Rn has countable chromatic number while the hypergraph of isosceles triangles in R2 does not, this for any number n>0.

Two graph games, with David Chodounsky, submitted to Electronic Journal of Combinatorics. We identify two determined games which can be used to prove that certain sigma-algebraic graphs have countable chromatic number.

Coloring closed Noetherian graphs, Journal of Mathematical Logic, in print. I show that for every algebraic graph on a Euclidean space, either it has a perfect clique or it is consistent with ZF+DC that it has countable chromatic number and Vitali set does not exist.

Coloring the distance graphs Eur. J. Math. 9 (2023), no. 3, 66. For every n>0, let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. It is consistent with ZF+DC that the chromatic number of Gn is countable while the chromatic number of Gn+1 is not.

Coloring triangles and rectangles   accepted to Commentationes Mathematicae Universitatis Carolinae. It is consistent with ZF+DC that the chromatic number of the rectangle hypergraph in n dimensions is countable while the chromatic number of equilateral triangles in two dimensions is uncountable.

Krull dimension in set theory  Ann. Pure Appl. Logic 174 (2023), no. 9, Paper No. 103299, 16 pp. For every n>0, let Dn be the hypergraph of all rectangles on n-dimensional Euclidean space. It is consistent with ZF+DC that the chromatic number of Gn is countable while the chromatic number of Gn+1 is not.

Polar forcings and measured extensions, with Paul Larson. Topology Appl. 323 (2023), Paper No. 108290, 12 pp. A sigma-ideal on a Polish space is polar if it is the intersection of null ideals of some collection of measures. We analyze the quotient forcings obtained by such ideals and prove iteration and preservation theorems for them.

Sequential topologies and Dedekind finite sets MLQ Math. Log. Q. 68 (2022), no. 1, 107–109. It is consistent with ZF that the topology of real numbers is not sequential while every infinite set of reals contains a countable infinite subset.

Coloring the distance graphs in three dimensions submitted to Combinatorica. For every n>0, let Gn be the graph on n-dimensional Euclidean space connecting points of rational distance. It is consistent with ZF+DC that the chromatic number of G3 is countable while the chromatic number of G4 is not.

Transcendental pairs of generic extensions

Noetherian spaces in choiceless set theory It is consistent with ZF+DC that every K sigma Polish field has a transcendence basis over a countable subfield, yet there is no nonprincipal ultrafilter over the natural numbers, the Lebesgue null ideal is closed under well-ordered unions etc. There are many independence results obtained by the same method.

Subadditive families of hypergraphs Accepted to Annals of Pure and Applied Logic. For proper forcings defined from a family of hypergraphs on a Polish space, I provide a simple combinatorial criterion on the hypergraphs equivalent to the forcing not adding an independent real.

Geometric set theory, with Paul Larson. AMS Surveys and Monographs 248

Hypergraphs and proper forcing Journal of Mathematical Logic Vol. 19, No. 02, 1950007 (2019) For a countable family of analytic hypergraphs on a Polish space, consider the sigma-ideal generated by Borel sets which are anticliques with respect to one of them. It turns out that many forcings can be obtained as quotient forcings of such sigma-ideals, there is a close connection between combinatorial properties of the hypergraphs and preservation properties of the forcings, and one can prove suitable iteration and product forcing theorems.

Namba forcing axiom may fail, Mathematical Logic Quarterly (2018), https://doi.org/10.1002/malq.201700025

Cardinal invariants of closed graphs, with Francis Adams, Israel Journal of Mathematics, (2018), https://doi.org/10.1007/s11856-018-1745-6

Strong measure zero sets in Polish groups, with Michael Hrusak, Illinois Journal of Mathematics 60 (2016), Number 3-4, 751-760

Canonical models for fragments of the axiom of choice, with Paul Larson, Journal of Symbolic Logic 82 (2017) 2, 489-509

Ramsey ultrafilters and countable-to-one uniformization, with Richard Ketchersid and Paul Larson, Topology and Its Applications 213 (2016), 190–198

Interpreter for topologists, Journal of Logic and Analysis 7 (2015), 1-61

Why Y-c.c., with David Chodounsky, Annals of Pure and Applied Logic 166 (2015) 1123-1149

Dimension theory and forcing, Topology and Its Applications 167 (2014) 31-35

Cofinalities of Borel ideals, with Michael Hrusak and Diego Rojas Rebolledo, Mathematical Logic Quarterly 1-9 (2014) DOI 101002/malq.201200079

Separation problems and forcing. J. Math. Log. 13 (2013), no. 1, 1350002

Canonical Ramsey theory on Polish spaces, with Marcin Sabok and Vladimir Kanovei, Cambridge Tracts in Mathematics 202, Cambridge University Press 2013, ISBN 978-1-107-02685-8

On the Steinhaus and Bergman properties for infinite products of finite groups, with Simon Thomas,  Confluentes Math. 4 (2012), no. 2, 1250002

Pinned equivalence relations, Mathematical Research Letters 18 (2011) 559-564

More ideals with the Komjath-Laczkovich property, Topology and Its Applications 158 (2011) 1149-1156

Forcing properties of ideals of closed sets, with Marcin Sabok, J. Symbolic Logic 76 (2011) 1075–1095

Ramsey theorem for product of finite sets with submeasures, with Saharon Shelah, Combinatorica 31 (2011) 225-244

On the existence of a sigma-closed dense subset, Comment.Math.Univ.Carolin. 51,3 (2010) 513-517

Applications of the ergodic iteration theorem. Math. Log. Q. 56 (2010), no. 2, 116-125

Regular embeddings of the stationary tower and Woodin’s ∑22 maximality theorem, with Richard Ketchersid and Paul Larson, J. Symbolic Logic 75 (2010), no. 2, 711-727

Preserving P-points in definable forcing. Fund. Math. 204 (2009), no. 2, 145-154

Increasing Δ12 by a Namba-style forcing, with Richard Ketchersid and Paul Larson, J. Symbolic Logic 72 (2007), 1372–1378

On the structure of stationary sets, with Qi Feng and Thomas Jech, Sci. China Ser. A 50 (2007) 615-627

Forcing with quotients, with Michael Hrusak, Archive Math. Logic 47 (2008), 719-739

Forcing idealized, Cambridge Tracts in Mathematics 174, Cambridge University Press 2008, ISBN 9780521874267

Proper forcing and rectangular Ramsey theorems, Israel J. Math. 152 (2006), 29–47

Between Maharam’s and von Neumann’s problem, with Ilijas Farah, Math. Research Letters 11 (2004), 673–684

Four and more, Ann. Pure Appl. Logic, with Ilijas Farah, Ann. Pure Appl. Logic 140 (2006), 3–39

Descriptive set theory and definable forcing, Memoirs Amer. Math. Soc. 793 (2004)

Games with creatures, with S. Shelah, Comm. Math. Univ. Carolinae 44 (2003), 9–23

Duality and the PCF theory, with S. Shelah, Math. Research Letters 9 (2002), 585–595

Forcing with ideals of closed sets, Comm. Math. Univ. Carolinae 43,1 (2002), 181–188

Isolating cardinal invariants, J. Math. Logic, 2003, 143-162

Terminal notions in set theory, Ann. Pure Appl. Logic 109 (2001), 89–116

Transfinite open games, Topology and Its Applications 111 (2001), 289–297

Killing ideals and adding reals, J. Symbolic Logic 65 (2000), 747–755

The nonstationary ideal and the other sigma ideals on omega one, Trans. Amer. Math. Soc. 352 (2000), 3981–3993

Terminal notions, Bull. Symbolic Logic 5 (1999), 470–484

On the Alaoglu-Birkhoff equivalence of posets, with S. Todorcevic, Illinois J. Math. 43 (1999), 281–292

Canonical models for aleph one combinatorics, with S. Shelah, Ann. Pure Appl. Logic 98 (1999), 217–259

Proper forcing and absoluteness in L(R), with I. Neeman, Comm. Math. Univ. Carolinae 39 (1998), 281–301

A dichotomy for forcing notions, Math. Res. Lett. 5 (1998) 213–226

Preserving sigma-ideals, J. Symbolic Logic 63 (1998), 1437–1441

Keeping additivity of the null ideal small, Proc. Amer. Math. Soc. 125 (1997), 2443–2451

Embeddings of Cohen algebras, with S. Shelah, Adv. Math. 126 (1997), 93–119

Semi-Cohen boolean algebras, with B. Balcar and T. Jech, Ann. Pure Appl. Logic 87 (1997), 187–208

Strongly almost disjoint functions, Israel J. Math. 97 (1997), 101–111

Small forcings and Cohen reals, J. Symbolic Logic 62 (1997), 280–284

Splitting number at uncountable cardinals, J. Symbolic Logic 62 (1997), 35–42

A classification of definable partial orders on omega one, Fund. Math. 153 (1997), 141-144

Characterization of the club forcing, in Papers on General Topology and Applications, S. Andima, R. Flagg, G. Itzkowitz,

Y. Kong, R. Kopperman amd P. Misra, eds., Annals of the New York Academy of Sciences 806 (1996), 476–484

A new proof of Kunen inconsistency, Proc. Amer. Math. Soc. 124 (1996), 2203-2205

More on the cut and choose game, Ann. Pure Appl. Logic 76 (1995), 291–301