The generalized continuum hypothesis revisited

We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering the known independence results, is “for most pairs κ<λ of regular cardinals, λ to the revised power of κ is equal to λ”. What is the revised power? λ to the revised power of κ is the minimal cardinality of a family of subsets of λ each of cardinality κ such that any other subset of λ of cardinality κ is included in the union of strictly less than κ members of the family. We still have to say what “for most” means. The interpretation we choose is: for every λ, for every large enoughK < ? _w . Under this reinterpretation, we prove the Generalized Continuum Hypothesis.     

Universal Graphs with Forbidden Subgraphs and Algebraic Closure

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated “algebraic closure” operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.     

Atomic saturation of reduced powers

Our aim was to try to generalize some theorems about the saturation of ultrapowers to reduced powers. Naturally, we deal with saturation for types consisting of atomic formulas. We succeed to generalize “the theory of dense linear order (or T with the strict order property) is maximal and so is any $\mathrm{pair}(T,\mathrm{\Delta })$ which is SOP₃”, (where Δ consists of atomic or conjunction of atomic formulas). However, the theorem on “it is enough to deal with symmetric pre-cuts” (so the $\mathfrak{p}=\mathfrak{t}$ theorem) cannot be generalized in this case. Similarly the uniqueness of the dual cofinality fails in this context.     

On thep-rank of Ext

AssumeV =L andλ is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext (G, ℤ) (under well known limitations), we construct an abelian groupG of cardinalityλ such that for noG′ ⊆G, |G′| <λ isG/G′ free and Ext (G, ℤ) realizes our requirements. Deceased. Partially supported by NSERC. Partially supported by the United States — Israel Binational Science Foundation. Publication No. 314.     

Possible size of an ultrapower of \omega

Let be the first infinite ordinal (or the set of all natural numbers) with the usual order . In § 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of , whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [1], modulo the assumption of supercompactness. In § 2 we construct several -Archimedean ultrapowers of under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a -Archimedean ultrapower of for some uncountable cardinal . This answers a question in [8], modulo the assumption of measurability. Received: 19 November 1996     

A parallel to the null ideal for inaccessible $$\lambda $$: Part I

It is well known how to generalize the meagre ideal replacing \(\aleph _0\) by a (regular) cardinal \(\lambda > \aleph _0\) and requiring the ideal to be \(({<}\lambda )\)-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing \(\aleph _0\) by \(\lambda \). So naturally, to call it a generalization we require it to be \(({<}\lambda )\)-complete and \(\lambda ^+\)-c.c. and more. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of \(\lambda \)-Borel sets modulo the ideal. Common wisdom have said that there is no such thing because we have no parallel of Lebesgue integral, but here surprisingly first we get a positive \(=\) existence answer for a generalization of the null ideal for a “mild” large cardinal \(\lambda \)—a weakly compact one. Second, we try to show that this together with the meagre ideal (for \(\lambda \)) behaves as in the countable case. In particular, we consider the classical Cichoń diagram, which compares several cardinal characterizations of those ideals. We shall deal with other cardinals, and with more properties of related forcing notions in subsequent papers (Shelah in The null ideal for uncountable cardinals; Iterations adding no \(\lambda \)-Cohen; Random \(\lambda \)-reals for inaccessible continued; Creature iteration for inaccesibles. Preprint; Bounding forcing with chain conditions for uncountable cardinals) and Cohen and Shelah (On a parallel of random real forcing for inaccessible cardinals. arXiv:1603.08362 [math.LO]) and a joint work with Baumhauer and Goldstern.     

Groups isomorphic to all their non-trivial normal subgroups

A finite collectionP of finite sets tiles the integers iff the integers can be expressed as a disjoint union of translates of members ofP. We associate with such a tiling a doubly infinite sequence with entries fromP. The set of all such sequences is a sofic system, called a tiling system. We show that, up to powers of the shift, every shift of finite type can be realized as a tiling system. Some of this work was done at the Mathematical Sciences Research Institute (MSRI), where research is supported in part by NSF grant DMS-9701755. The first two authors thank K. Schmidt for useful conversations and ideas.     

Finite subgraphs of uncountably chromatic graphs

It is consistent that for every function f:ω → ω there is a graph with size and chromatic number ?₁ in which every n‐chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erd?s. It is consistent that there is a graph X with Chr(X)=|X|=?₁ such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)≤?₂ (so the Taylor conjecture may fail). It is also consistent that if X is a graph with chromatic number at least ?₂ then for every cardinal λ there exists a graph Y with Chr(Y)≥λ all whose finite subgraphs are induced subgraphs of X. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 28–38, 2005     

More Jonsson Algebras

 We prove that on many inaccessible cardinals there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal λ is λ × ω-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra of cardinality . Lastly, we give further information on guessing of clubs. Received: 10 March 1992 / First revised version: 11 August 1997 / Second revised version: 12 September 2000 / Published online: 5 November 2002