Poset algebras over well quasi-ordered posets.
with Robert Bonnet and Wieslaw Kubis. Algebra Universalis http://dx.doi.org/10.1007/s00012-008-2063-3A note on the Engelking-Karlowicz theorem.
with Yimu Yin. Acta Mathematica Hungarica. 2008. ( pdf.)We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Kar{\l}owicz (in \cite{EK}). In these graphs, the vertices are subsets of an ordinal, and two subsets $X$ and $Y$ are connected iff for some $a\in X\cap Y$ the order type of $a\cap X$ is different from that of $a\cap Y$. In addition to the chromatic number $\chi(G)$ of these graphs we study $\chi_\kappa(G)$, the $\kappa$-chromatic number, which is the least cardinal $\mu$ with a decomposition of the vertices into $\mu$ classes none of which contains a $\kappa$-complete subgraph.
Some results in polychromatic Ramsey theory.
with James Cummings and Clifford Smyth. J. Symbolic Logic 72, 3 (2007), 865--896. ( pdf.)Infinite games on finite sets.
with Rene Schipperus. Israel J. of Math. 159 (2007) 205--219. ( pdf.)On Jakovlev spaces.
with Isac Gorelic and Istvan Juhasz. Israel Journal of Mathematics 152 (2006) 205--219. ( pdf.)A topological space $X$ is called weakly first countable, if for every point $x$ there is a countable family $\{C_n^x \mid n\in\omega\}$ such that $x\in C_{n+1}^x \subseteq C_n^x $ and such that $U \subset X$ is open iff for each $x \in U$ some $C_n^x$ is contained in $U$. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N.N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V.I.Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption $\gb = \gc$, and also by forcing.
Covering $R^{n+1}$ by graphs of n-ary functions and long linear orderings of Turing degrees.
with Stefan Geschke. Uri Abraham, Stefan Geschke, Proc. Amer. Math. Soc. 132 (2004), 3367-3377. ( dvi, 10pp)A point $(x_0,\dots,x_n)\in X^{n+1}$ is covered by a function $f:X^n\to X$ iff there is a permutation $\sigma$ of $n+1$ such that $x_{\sigma(0)}=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$. By a theorem of Kuratowski \cite{kuratowski}, for every infinite cardinal $\kappa$ exactly $\kappa$ $n$-ary functions are needed to cover all of $(\kappa^{+n})^{n+1}$. We show that for arbitrarily large uncountable $\kappa$ it is consistent that the size of the continuum is $\kappa^{+n}$ and $\mathbb R^{n+1}$ is covered by $\kappa$ $n$-ary continuous functions. We study other cardinal invariants of the $\sigma$-ideal on $\mathbb R^{n+1}$ generated by continuous $n$-ary functions and finally relate the question of how many continuous functions are necessary to cover $\mathbb R^2$ to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.
Ladder gaps over stationary sets.
with S. Shelah. J. Symbolic Logic 69 (2004), no 2, 518-532. ( dvi 21 ppFor a stationary set $S\subseteq \omega_1$ and a ladder system $C$ over $S$, a new type of gaps called $C$-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set $S$, for every ladder $C$ over $S$, every gap contains a subgap that is $C$-Hausdorff. But for every ladder $E$ over $\omega_1\setminus S$ there exists a gap with no subgap that is $E$-Hausdorff. A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c posets is again a polarized c.c.c poset.
Lusin sequences under CH and under Martin's Axiom.
with Saharon Shelah.Fundamenta Mathematicae 169 (2001), pp 97-103. ( ps 8 pp).
Assuming the continuum hypothesis there is an inseparable sequence of length $\omega_1$ that contains no Lusin subsequence, while if Martin's Axiom and $\neg CH$ is assumed then every inseparable sequence (of lengt$\omega_1$) is a union of countably many Lusin subsequences.
Hausdorff's theorem for posets that satisfy the finite antichain property.
with Robert Bonnet.Fundamenta Mathematicae, 159 (1999) pp. 51--69. ( ps 24 pp).
Martin's axiom and $\Delta^2_1$ well-ordering of the reals
with Saharon Shelah.Archive for Mathematical Logic, 35 (1996) 287--298. ( compressed ps 16). (a link to Springer's Archive for Mathematical Logic version)
Assuming an inaccessible cardinal $\kapa$, there is a generic extension in which $MA + 2^{\aleph_0} = \kapa$ holds and the reals have a $\Delta^2_2$ well-ordering.
See also Coding with Ladders for an extension which does not require an inaccessible cardinal.
A \Delta^2_2 Well-Order of the Reals And Incompactness of L(Q^{MM})
with Saharon Shelah.Annals of Pure and Applied Logic 59 (1993) 1--32. ( ps 49 pp).
A forcing poset of size $2^{2^{\aleph_1}}$ which adds no new reals is described and shown to provide a $\Delta^2_2$ definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: Some products are special while other are Suslin trees.
The paper also deals with the Magidor-Malitz logic: it is consistent that this logic is highly non compact.
Cardinal Arithmetic
with Menachem Magidor.in: Handbook of Set Theory. Foreman, Kanamori, and Magidor (eds.) ( dvi, 65 pp).
An introduction, mainly to Shelah's pcf theory.
Proper forcing.
in: Handbook of Set Theory. Foreman, Kanamori, and Magidor (eds.) ( pdf 72 pp).An introduction to Shelah's theory of countable support iteration of Proper Forcing.
On a superatomic Booean algebra which is not generated by a well-founded sublattice.
with Matatyahu Rubin and Robert Bonnet.Israel. J. Math., Vol. 123, 2001, 221-239. ( compressed ps 24 pp).
Let \b denote the unboundedness number of \omega^{\omega}. That is, \b is the smallest cardinality of a subset F \subseteq \omega^{\omega} such that for every g \in \omega^{\omega} there is f \in F such that \{ n \mid g(n) \leq f(n)\} is infinite.
A Boolean algebra B is well-generated, if it has a well-founded sublattice L such that L generates B. We show that it is consistent with ZFC that \aleph_1 < 2^{\aleph_0} = \b, and there is a Boolean algebra B such that B is not well-generated, and B is superatomic with cardinal sequence \langle \aleph_0, \aleph_1, \aleph_1, 1 \rangle.
This result is motivated by the fact that if the cardinal sequence of a Boolean algebra $B$ is $\langle \aleph_0, \aleph_0, \lambda, 1 \rangle$, and $B$ is not well-generated, then $\lambda \geq \mbox{\sgoth b}$.
On poset boolean algebras.
with Robert Bonnet Wielslaw Kubis and Matatyahu Rubin.Order, 20 (2003), no. 3, 265--290. ( pdf 43 pp).
If P is a poset then F(P) denotes the boolean algebra generated by P. A Boolean algebra is well-generated if it is generated by a sublatice that is well-founded. Such algebras are superatomic. The first main theorem in the paper is the following. Suppose that every antichain in P is finite and the rationals do not embbed into P. Then F(P) is well-generated.
The second main theorem is the following. If every antichain of P is finite and the rationals do not embbed into P, then F(P) is a homomorphic image of a subalgebra of F(W) where W is a (partially) well-ordered poset.
Partition Properties of $\omega_1$ Compatible with CH.
with Stevo Todorcevic.Fundamenta Mathematicae 152 (1997) pp 165--180. ( ps 17 pp).
A combinatorial statement concerning ideals of countable subsets of $\omega_1$ is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Souslin Hypothesis, that all $(\omega_1, \omega_1^*)$-gaps are Hausdorff, and that every coherent sequence on $\omega_1$ either almost includes or is orthogonal to some uncountable subset of $\omega_1$.
An earlier version of this paper is ( ps 14 pp).
Coding with ladders a well-ordering of the reals.
with Saharon Shelah.The Journal of Symbolic Logic, 67, Number 2 (2002) 579--597. ( compressed dvi, gz 30 pp).
Any model of ZFC + GCH has a generic extension (made with a poset of size $\aleph_2$) in which the following hold: $MA + 2^{\aleph_0} = \aleph_2 +$ {\em there exists a} $\Delta^2_1$-{\em well ordering of the reals.} Since the model is obtained by a small forcing, it retains all large cardinal properties of the ground model. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on $\omega_1$. Therefore, the study of such ladders is a main concern of this article.
Compared with Martin's axiom and \Delta^2_1 well-ordering of the reals no inaccessible cardinal is need here (but the continuum is only aleph-two).
The paper was written at York University, Canada, and I wish to thank my hosts at the Department of Mathematics and Statistics.
The following older papers have no electronic version available, but you can ask for reprints.
Free sets for commutative families of functions
in: Logic, Methodology and Philosophy of Science VIII J. E. Fenstad et al., eds. Studies of Logic and the Foundations of Mathematics, Volume 126, North-Holland. Elsevier Science Publishers (1989) pp 295-212.Since this was a plenary lecture attended by scientists from different fields, I choose to expressed my views on the import of consistency proofs. The paper also contains a short proof showing that SOCA implies that any commutative family of functions on $\omega_1$ has an uncountable free set.
On the consistency of some partition theorems for continuous colorings, and the structure of aleph-one -dense real order types.
with Matatyahu Rubin and Saharon Shelah.Annals of Pure and Applied Logic 29 (1985) 123--206. Because of its length the paper's ps file is divided into two parts
(pdf photocopy of the paper part 1) (pdf photocopy of the paper part 2).
We present some techniques in c.c.c forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of aleph-one-dense sets of real numbers. In this direction we continue the work of Baumgartner who proved the axiom BA stating that every two aleph-one-dense homogeneous subsets of R are isomorphic, is consistent. We prove Con( BA + continuum is above aleph two). Let K be the set of order types of aleph-one-dense homogeneous subsets of R with the relation of embeddability. We prove for every finite ordering L that MA + K is isomorphic to L is consistent iff L is a distributive lattice. We formulate, prove the consistency, and show implications of different open coloring axioms.
Isomorphism types of Aronszajn trees.
with Saharon Shelah.Israel Journal of Mathematics, vol. 50 (1985) pp 75--113.
We study the isomorphism types of Aronszajn trees of height \omega_1, and give divers results on this question (mainly consistency results). The main theorem perhaps is the consistency of "Every two Aronszajn trees are isomorphic on a club set (and Martin's Axiom)". This is from my Phd thesis.
A minimal model for non-CH: iteration of Jensen's reals.
Transactions of the American Mathematical Society, Vol. 281 No 2 (1984) pp. 657--674.A model of ZFC + 2^{\aleph_0} = \aleph_2 is constructed which is minimal with respect to being a model of non-CH. Any strictly included submodel of ZF (which contains all the ordinals) satisfies CH. In this model the degrees of constructibility have order type \omega_2. A novel method of using the diamond is aplied here to construct a countable-support iteration of Jensen's reals: in defining the \alph's stage of the iteration the diamond "guesses" possible \beta > \alpha stages of the iteration.
Construction of a rigid Aronszajn tree
Proceedings of the American Mathematical Society, Vol. 77, No 1 (1979) pp. 136--137.We construct in ZFC an Aronszajn tree with no automorphism.
On the intersection of closed unbounded sets
with Saharon Shelah.The journal of Symbolic Logic Vol. 51, No 1 (1986) pp. 180--189.
A theorem of Galvin says, for example, that if CH holds then for any family
of \aleph_2 club subsets of \omega_1 there is a club contained in
unbountably many members of that family. In contrast we have the
following example theorems.
Theorem 1.1 (Abraham) Assume CH.
There is a generic extension which
does not add new countable sequences, collapses no cardinals, and in which
2^{(\kappa^+)}=\lambda and the following holds:
Theorem INCOMPLETE
Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra.
with Robert Bonnet.( pdf 30 pp).
Proceedings of the American Mathematical Society Vol. 115, No3 (1992) pp 585--592.
A Boolean algebra is superatomic iff every subalgebra is atomic. The boolean algebra generated by the intervals [a, b) of a chain is called the interval algebra. The interval algebra of an ordinal is superatomic, and the theorem thus says that no other superatomic algebras can be obtaind by considering subalgebras of non well-founded interval algebras.
Martin's Axiom does not imply that every two \aleph_1 dense sets of reals are isomorphic
with Saharon Shelah.Israel Journal of Mathematics. Vol. 38, No. 1-2 (1981) pp. 161--176.
Assuming the consistency of ZFC we prove the claim in the title by showing the consistency with ZFC of: There exists a set of reals A such that every function from A to A is order preserving on an uncountable set. We prove related results among which is the consistency with ZFC of: Every function from the reals into the reals is monotonic on an uncoutnable set.
A note on Dilworth's Theorem in the infinite case.
Order Vol. 4 (1987) pp 107--125.If every antichain of a poset is finite, then the set of antichains is well-founded (under inverse inclusion). We compute the rank of this antichain posets for the poset \alpha \times \beta where \alpha and \beta are ordinals. The main theorem we prove in the paper is that if the rank of the well-founded poset of antichains is less than $\omega_1^2$ then the poset is the union of countable many chains.
Squares with diamonds and Souslin trees with special squares.
with Saharon Shelah and Robert M. Solvay.Fundamenta Mathematicae, Vol. 127 (1986) pp 133--162.
The squares and the diamonds are useful set-theoretic axioms used in construction of infinite objects. Here we introduce and study different versions of such combinatorial principles on successor of singular cardinals. We prove some implications (in ZFC) inquire the situation in $L$, and give an application (the construction of a Souslin tree on a successor of a singular cardinal whose square is special).
The consistency with CH of some consequences of Martin's Axiom plus 2^{\aleph_0} > \aleph_1.
with Keith J. Devlin and Saharon Shelah.Israel Journal of Mathematics. Vol. 31 No 1 (1978) pp 19--33.
We present a weak axioma(called SAD--this is a joke of Devlin's) that implies some of the consequences of MA, but is consistent with GCH. We use the method of Jensen in his proof of the consistency with GCH of the Suslin Hypothesis.
For example, SAD implies that every Hajnal--Mate graph on \omega_1 has a countable chromatic number.
The degrees of constructibility of Choen reals.
with Richard A. Shore.Proc. London Math. Soc. (3), 53 (1986) pp. 193--208.
The paper studies the structure of the reals after adding a cohen real, under the relation x \in L[y].
Forcing with stable posets
with Saharon Shelah.The Journal of Symbolic Logic, Vol.47, No 1 (1982) pp 37--42.
A weak form of Martin's Axiom is presented (which deals with all stable posets). In the extension 2^{\aleph_0}>\aleph_1 holds, but some consequences of CH still hold.
Initial segments of the degrees of size \aleph_1.
with Richard A. Shore.Israel Journal of MAthematics, Vol. 51 No 1 (1986) pp 1--51.
We settle a series of questions first raised by Yates at the Jerusalem (1968) Colloquium on Mathematical Logic by characterizing the initial segments of the degrees of unsolvability of size \aleph_1: Every upper semi-lattice of size \aleph_1 with zero, in which every element has at most countably many predecessors, is isomorphic to an initial segment of the Turing degrees.
Martin's Axiom and first-countable S- and L-spaces.
with Stevo Todorcevic.in: K. Kunen and J. E. Vaughan (Eds.) Handbook of Set-Theoretic Topology, pp 327--346, Elsevier Science Publishers (1984).
The main result proved there is the consistency of Martin's Axiom + non-CH with the existence of a first-countable S-space
Aronszajn trees on \aleph_2 and \aleph_3.
Annals of Pure and Applied Logic Vol. 24 (1983) pp 213--230.Assuming the existence of a supercompact cardinal and a weakly compact cardinal above it, we provide a generic extension where there are no Aronszajn trees of height \omega_2 or \omega_3. On the other hand (this is due to M. Magidor) if there are no Aronszajn trees on aleph two and three then O^# exists.
Minimal Model of "\aleph_1^L is countable" and definable reals.
Advances in Mathematics, Vol. 55 No 1 (1985) pp 75--89.
There is a generic extension M[a], where a \subseteq \omega is such that:
1. M[a] is a minimal model of "\aleph_1^L is countable".
2. a is a \Pi_2^1 singleton in M[a] and all the constructible reasl are recursive in
a.
Free sets for nowhere-dense set mappings
Israel Journal of Mathematics Vol.39 (1981) pp 167--176.We give some consistency results on the existence of uncountable free sets for nowhere-dense set mappings. For example, we prove that it is relatively consistent with ZFC and MArtin's Axiom that any nowher-dense set maping defined on the reals has an uncountable free set.
On forcing without the continuum hypothesis.
The Journal of Symbolic Logic Vol. 48 No. 3 (1983) pp 658--661.There is a poset R such that in V^R, \aleph_2 becomes of cardinality \aleph_1 but \aleph_1 and the cardinals above \aleph_2 are not collapsed. The problem is that CH may not hold--The idea is to force first Cohen reals.
Forcing closed unbounded sets
with Saharon Shelah.The Jounal of Symbolic Logic, Vol. 48 N0 3 (1983) pp 643--657.
We discuss that problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.
