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We force the existence of a non-trivial κ \kappa -complete ultrafilter over κ \kappa which fails to satisfy the Galvin property. This answers a question asked by Benhamou and Gitik [Ann. Pure Appl. Logic 173 (2022), Paper No. 103107].

We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.

There is a Turing functional $\Phi $ taking $A^\prime $ to a theory $T_A$ whose complexity is exactly that of the jump of A, and which has the property that $A \leq _T B$ if and only if $T_A \trianglelefteq T_B$ in Keisler’s order. In fact, by more elaborate means and related theories, we may keep the complexity at the level of A without using the jump.

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

We deal with the monadic (second-order) theory of order. We prove all known results in a unified way, show a general way of reduction, prove more results and show the limitation on extending them. We prove (CH) that the monadic theory of the real order is undecidable. Our methods are model-theoretic, and we do not use automaton theory. This is a slightly corrected version of a very old work.

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types., 32 pages, no figures

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

We prove that for lambda = beta_omega or just lambda strong limit singular of cofinality aleph_0, if there is a universal member in the class K^lf_lambda of locally finite groups of cardinality lambda, then there is a canonical one (parallel to special models for elementary classes, which is the replacement of universal homogeneous ones and saturated ones in cardinals lambda = lambda^ < lambda). For this, we rely on the existence of enough indecomposable such groups, as proved in "Density of indecomposable locally finite groups". We also more generally deal with the existence of universal members in general classes for such cardinals., arXiv admin note: text overlap with arXiv:1901.09747

The logic $${\mathbb{L}}_\theta ^1$$ was introduced in [She12]; it is the maximal logic below $${{\mathbb{L}}_{\theta, \theta}}$$ in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are $${\mathbb{L}}_\theta ^1$$ -equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic. Also for strong limit λ>θ of cofinality $${\aleph _0}$$ , every complete $${\mathbb{L}}_\theta ^1$$ -theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.

We show that it is independent whether club $\kappa $ -Miller forcing preserves $\kappa ^{++}$ . We show that under $\kappa ^{ \kappa $ , club $\kappa $ -Miller forcing collapses $\kappa ^{ to $\kappa $ . Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $ -Miller forcing does not have the Laver property.

We deal with the combinatorial principle Weak Diamond. We prove that, if it holds for a given cardinal, we can get this principle with more than two colours or some relevant ideal is not too saturated. Then we point out a model theoretic consequence of Weak Diamond.

A permutation group $G$ on a set $A$ is $��$-homogeneous iff for all $X,Y\in [A]^��$ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. $G$ is $��$-transitive iff for any injective function $f$ with $dom(f)\cup ran(f)\in [A]^{\le ��}$ and $|A\setminus dom(f)|=|A\setminus ran(f)|=|A|$ there is a $g\in G$ with $f\subset g$. Giving a partial answer to a question of P. M. Neumann we show that there is an $��$-homogeneous but not $��$-transitive permutation group on a cardinal $��$ provided (i) $����$ we give a method to construct large $��$-homogeneous, but not $��$-transitive permutation groups. Using this method we show that there exists $��^+$-homogeneous, but not $��^+$-transitive permutation groups on $��^{+n}$ for each infinite cardinal $��$ and natural number $n\ge 1$ provided $V=L$., 16 pages

We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.

We prove that it is consistent with large values of the continuum that there are no S-spaces. We also show that we can also have that compact separable spaces of countable tightness have cardinality at most the continuum.

We find new "reasons" for a class of models for not having a universal model in a cardinal $\lambda$. This work, though it has consequences in model theory, is really in combinatorial set theory. We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of $T_{\rm ceq}$ (essentially another representation of $T_{\rm feq}$ which was already considered but the proof with $T_{\rm ceq}$ is more transparent). Models of $T_{\rm ceq}$ consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in $\lambda$. This work is continued in [Sh:F2071]., Comment: 24 pages, after referee report

It was proved recently that Telgarsky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of $$\mathsf {GCH}+\square $$ . The proof introduces a combinatorial principle that is shown to follow from $$\mathsf {GCH}+\square $$ , namely: We prove this principle is independent of $$\mathsf {GCH}$$ and $$\mathsf {CH}$$ , in the sense that $$\bigtriangledown $$ does not imply $$\mathsf {CH}$$ , and $$\mathsf {GCH}$$ does not imply $$\bigtriangledown $$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $$\bigtriangledown $$ holds with $${\mathbb {P}}$$ equal to the weight- $$\aleph _\omega $$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $$\mathsf {ZFC}+\mathsf {GCH}$$ .

We prove that $$\mathfrak{mcf}$$ and $$\mathfrak{mcf}$$ are both consistent relative to ZFC. This answers a question by Banakh, Repovs and Zdomskyy and a question from [MS11].

Our aim was to generalize some theorems about the saturation of ultra-powers to reduced powers. Naturally, we deal via saturation for types consisting of atomic formulas. We succeed to generalize the theory of dense linear is maximal and so is any pair (T, Delta) which is SOP_3 (Delta consists of atomic or conjunction of atomic formulas). However, SOP_2 is not enough, so the p equals t theorem cannot be generalized in this case. Similarly the unique dual of cofinality.

If $$\kappa > \aleph_0$$ then $$\kappa \to(\kappa,\operatorname{Top}K_\kappa)^2$$ , i.e., every graph on $$\kappa$$ vertices contains either an independent set of $$\kappa$$ vertices, or a topological $$K_\kappa$$ , iff $$\kappa$$ is regular and there is no $$\kappa$$ -Suslin tree. Concerning the statement $$\omega_2\to(\operatorname{Top}K_{\omega_2})^2_\omega$$ , i.e., in every coloring of the edges of $$K_{\omega_2}$$ with countably many colors, there is a monochromatic topological $$K_{\omega_2}$$ , both the statement and its negation are consistent with the Generalized Continuum Hypothesis.

We prove that $ZF+DC+"$there exists a transcendence basis for the reals$"+"$there is no well-ordering of the reals$"$ is consistent relative to $ZFC$. This answers a question of Larson and Zapletal., We corrected a gap in the proof of Claim 6

We construct, in $\mathsf{ZFC}$, a countably compact subgroup of $2^{\mathfrak{c}}$ without non-trivial convergent sequences, answering an old problem of van Douwen. As a consequence we also prove the existence of two countably compact groups $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ such that the product $\mathbb{G}_{0} \times \mathbb{G}_{1}$ is not countably compact, thus answering a classical problem of Comfort., 21 pages, to be published in Transactions of the American Mathematical Society

Strongly bounded groups are those groups for which every action by isometries on a metric space has orbits of finite diameter. Many groups have been shown to have this property, and all the known infinite examples so far have cardinality at least $2^{\aleph_0}$. We produce examples of strongly bounded groups of many cardinalities, including $\aleph_1$, answering a question of Yves de Cornulier [4]. In fact, any infinite group embeds as a subgroup of a strongly bounded group which is, at most, two cardinalities larger.

We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no nontrivial homomorphism to ℤ (known classically for ℵ1-free, recently for ℵn-free). We succeed to prove the existence of even $${\aleph _{{\omega _1} \cdot n}}$$ -free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every $${\aleph _{{\omega _1} \cdot \omega }}$$ -free Abelian group has non-trivial homomorphisms to ℤ.

An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey’s Theorem asserting that in any coloring of the edges of a complete graph there exist large highly connected subgraphs all of whose edges are colored by the same color.

We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.

We prove the consistency of: for suitable strongly inaccessible cardinal λ \lambda the dominating number, i.e., the cofinality of λ λ {}^\lambda \lambda , is strictly bigger than cov λ _\lambda (meagre), i.e., the minimal number of nowhere dense subsets of λ 2 {}^\lambda 2 needed to cover it. This answers a question of Matet.

We prove that if the sequence $$\langle k_n:1 \le n < \omega\rangle$$ contains a so-called gap then the sequence $$\langle S^{\aleph_n}_{\aleph_{k_n}}:1 \le n < \omega\rangle$$ of stationary sets is not mutually stationary, provided that $$k_n

We consider a version of $$\kappa $$ κ -Miller forcing on an uncountable cardinal $$\kappa $$ κ . We show that under $$2^{ 2 < κ = κ this forcing collapses $$2^\kappa $$ 2 κ to $$\omega $$ ω and adds a $$\kappa $$ κ -Cohen real. The same holds under the weaker assumptions that $${{\,\mathrm{cf}\,}}(\kappa ) > \omega $$ cf ( κ ) > ω , $$2^{2^{ 2 2 < κ = 2 κ , and forcing with $$([\kappa ]^\kappa , \subseteq )$$ ( [ κ ] κ , ⊆ ) collapses $$2^\kappa $$ 2 κ to $$\omega $$ ω .