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# Papers

 PFA AND THE NONSTATIONARY IDEAL, Proceedings of the American Math Society, vol. 146, (2018), pp. 791-802. Abstract We adapt the method of Neeman of forcing with side conditions to show that PFA does not imply the precipitousness of the nonstationary ideal on $$\omega_1$$ 

 A DIRECT PROOF OF THE FIVE ELEMENT BASIS THEOREM, with G. Venturi, Mathematical Logic Quarterly, vol. 63(3-4), (2017), pp. 289-298 Abstract We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of Larson, Koenig, Moore and Velickovic and simplifilesthe original proof of Moore. 

 POSITIONAL STRATEGIES IN LONG EHRENFEUCHT-FRAISSE GAMES, with S. Shelah and J. Väänänen Journal of Symbolic Logic vol.85 (1), (2015), 285-300. Abstract We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fraissé-game of length \omega_1 but there is no \sigma-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist. 

 STATIONARY SET REFLECTION AND TWO CARDINAL TREE PROPERTIES, with H. Sakai, J. Inst. Math. Jussieu, vol.14 (1), (2015), 69–85. Abstract We study consequences of stationary and semi-stationary set refl ection. We show that the semi-stationary re ection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi-stationary set refl ection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two refl ection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set reflection is analogous to strong compactness. 

 PCF STRUCTURES OF HEIGHT LESS THAN $$\omega_3$$, with K. Er-rhaimini, Journal of Symbolic Logic vol.75 (4), (2010), no. 4, 1231–1248. 

 FORCING AXIOMS AND CARDINAL ARITHMETIC, Logic Colloquium 2006, Lect. Notes in Logic, Association for Symbolic Logic, Chicago, IL, (2009), pp. 328--360;   MAHARAM ALGEBRAS, Ann. Pure Appl. Logic, vol. 158 (3), (2009), pp. 190--202;   BOUNDING THE CONSISTENCY STRENGTH OF A FIVE ELEMENT LINEAR BASIS, with B. Koenig, P. Larson and J. T. Moore , Israel Journal of Mathematics, vol. 164 (1), (2008), pp. 1-18;   MAHARAM ALGEBRAS AND COHEN REALS, with I. Farah, Proceedings of the Amer. Math. Society, vol. 135 (7), (2007), pp. 2283-2290;   BOUNDED PROPER FORCING AXIOM AND WELL ORDERINGS OF THE REALS, with A. Caicedo, Mathematical Research Letters, vol. 13 (2-3), (2006), pp. 393--408;   Von Neumann's Problem and Large Cardinals, with I. Farah, Bulletin of the London Mathematical Society, vol.38 (6), (2006), pp. 907--912;   CCC Forcing and Splitting Reals, Israel Journal of Mathematics, vol. 147, (2005), pp. 209-220;   Collapsing functions, with E. Schimmerling, Mathematical Logic Quarterly, vol. 50, (2004), pp. 3-8;   Games played on partial isomorphisms, with J. Väänänen, Archive for Mathematical Logic, vol. 49 (1), (2004), pp. 19-30;   The basis problem for CCC posets, Proceedings of the Hajnal Conference, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 58, (2002), pp. 149-160;   On the complexity of the isomorphism relation for fields of finite transcendence degree, with S. Thomas, Journal of Pure and Applied Algebra, vol. 159 (2-3), (2001), pp. 347-363;   Asymptotic cones of finitely generated groups, with S. Thomas, Bulletin of the London Mathematical Society, vol. 32, (2000), pp. 203-220;   Analytic ideals and cofinal types, with A. Louveau, Annals of Pure and Applied Logic, vol. 99 (1-3), pp. 171-195;   On the complexity of the isomorphism relation for finitely generated groups, with S. Thomas, Journal of Algebra, vol. 217 (1), pp. 352-373;   A note on Tsirelson type ideals, Fundamenta Mathematicae, vol. 159 (3), (1999), pp. 259-268;   Complexity of reals in inner models of set theory, with H. Woodin, Annals of Pure and Applied Logic, vol. 92, (1998), pp. 283-295;   Nonstandard Models and Analytic Equivalence Relations, with S. Friedman, Proceedings of the American Mathematical Society, vol. 125, no. 6, (1997), pp. 1807-1809;   Delta_1-Definability, with S. Friedman, Annals of Pure and Applied Logic, vol. 89, no. 1, pp. 93-99;   Hereditary normality of yN, with P. Nyikos and L. Soukup, Topology and its applications, vol. 65, (1995), pp. 9-19;   Borel partitions of infinite subtrees of a perfect tree, with A. Louveau and S. Shelah, Annals of Pure and Applied Logic, vol. 63, (1993), pp. 271-281;   Complete normality and countable compactness, with P. Nyikos, B. Shapirovskii and Z. Szentmiklossy, Topology Proceedings, vol.17 (1992), pp. 395-403;   A note on Borel equivalence relations, with A. Louveau, Proceedings of American Mathematical Society, vol. 120 (1), (1994), pp. 255-259;   Deterministic Approximate Counting of Depth-2 Circuits, with M. Luby, A. Wigderson, Proceedings of the Second Israeli Symposium on Theory of Computing and Systems, 1993, pp. 18-24;   OCA and automorphisms of P(N)/FIN, Topology and its applications, vol. 49, (1993), pp. 1-13;   Efficient Approximation of Product Distributions, with G. Even, O. Goldreich, M. Luby, N. Nisan, Random Structures and Algorithms, vol. 13, No. 1, pp. 1-16, August 1998 - (Journal version);   Approximations of General Independent Distributions, with G. Even, O. Goldreich, M. Luby, N. Nisan, Proceedings of the 24th Symposium on the Theory of Computing, (STOC), pp. 10-16, 1992. - (Conference version);   On Deterministic Approximation of DNF, with M. Luby, preliminary version in Prooceedings of the 23rd Symposium on the Theory of Computing, (STOC), (1991), pp. 430-438; final version in Algorithmica, vol. 16, no. 4/5, (Oct/Nov 1996), pp. 415-433;   Applications of the Open Coloring Axiom, W. Just et H. Woodin , eds. Springer Verlag, Berlin, (1992), pp. 137-154;   CCC posets of perfect trees, Compositio Mathematica, vol. 79, (1991), pp. 279-294;   Forcing axioms and stationary sets, Advances in Mathematics, vol. 94 (2), (1992), pp. 256-284;   Martin's axiom and partitions, with S. Todorcevic, Compositio Mathematica, vol. 63, (1987), pp. 391-407;   Playful Boolean algebras, Transactions of the American Mathematical Society, vol. 296 (2), (1986), pp. 727-740;   Jensen's box principle and the Novak number of partially ordered sets, Journal of Symbolic Logic, vol. 51 (1), (1986), pp. 47-58;   Definable automorphisms of P(N)/FIN, Proceedings of the American Mathematical Society, vol. 96 (1), (1986), pp. 130-135; 

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