On the number of non conjugate subgroups

We shall prove that every group of cardinality ₁ has at least ₁ non conjugate subgroups, and we shall generalize this theorem to many more uncountable cardinalities. For example underGCH for every uncountable cardinal and every groupG of cardinality ,G has at least non conjugate subgroups.Presented by W. Taylor.I would like to thank Rami Grossberg for writing and rewriting this paper, and Wilfrid Hodges for removing many errors and suggesting improvements in presentation; many facts are proved only due to his explicit request.This research was supported by grant (No. 1110) from the United States-Israel Binational Science Foundation.     

Constructions of many complicated uncountable structures and Boolean algebras

This article has three aims: (1) To make the results of [12, VIII] on constructing models more available for application, by separating the combinatorial parts. Thus in applications one will only need the relevant things from the area of application. (2) To strengthen the results there. In particular, we were mainly interested in [12, VIII] in showing that there are many isomorphism types of models of an unsuperstable theory, with results about the number of models not elementarily embeddable in each other being a side benefit. Here we consider the latter case in more detail, getting more cases. We also consider some more complicated constructions along the same lines % MathType!MTEF!2!1!+- . (3) To solve various problems from the list of van Dowen, Monk and Rubin [3] on Boolean algebras, which was presented at a conference on Boolean algebra in Oberwolfach January 1979 (most of the solutions are mentioned in the final version). Some of them are not related to (1) and (2). This continues [10, §2] in which the existence of a rigid B.A. in every uncountable power was proved. There (and also here) we want to demonstrate the usefulness of the methods developed in [12, VIII] (and §§ 1,2) for getting many (rigid) non-embeddable models in specific classes. The author thanks Don Monk for refereeing this paper very carefully, detecting many errors, adding many details, and shortening the proof of Theorem 4.1; and Rami Grossberg for carefully proofreading the paper. This research was partially supported by the United States-Israel Binational Science Foundation.     

Colouring without triangles and partition relation

We solved questions of Erdös and Hajnal and deal with related topics. In particular, we show (CH) that in an example ofN→[N] ₃ ² , there need not be a triangle with three colours.     

Whitehead groups may not be free even assuming ch,II

We prove some theorems on uncountable abelian groups, and consistency results promised in the first part, and also that a variant of called ♣ (club), is consistent with 2^{ℵ 0}<ℵ₁. The author would like to thank the United States-Israel Binational Science Foundation for Grant 1110 and the NSF for Grant MCS-08479.     

μ-complete Souslin trees on μ+

Summary We prove thatµ=µ ^<µ , 2 ^µ =µ ⁺ and there is a non-reflecting stationary subset ofµ ⁺ composed of ordinals of cofinality < imply that there is a -complete Souslin tree onµ ⁺.The second author thanks the Binational Science Foundation for supporting this research. Publication no. 449     

More on powers of singular cardinals

We give bounds for where cfδ=ℵ₁, (∀a<δ) , in cases which previously remained opened, including the first such cardinal: theω ₁-th cardinal inC _ω=∩n<ω C _n whereC ₀ is the cardinal andC _n+1 the set of fixed points ofC _n. No knowledge of earlier results is required. A subsequent work generalizing this was applied to many more cardinals ([Sh 7]). The author would like to thank the Canadian NSERC for supporting this research by Grant A3040 and the Israel Academy of Science for supporting it.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Majority decisions when abstention is possible

Suppose that we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential “voters.” The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in Shelah (2009) [7], but here we allow voters to abstain. Aside from the trivial case, the possible families of (partial) choice functions break into three cases in terms of the functions that can be generated by majority decision. In one of these, cycles along the lines of Condorcet’s paradox are avoided. In another, all partial choice functions can be represented.     

On the verge of inconsistency: Magidor cardinals and Magidor filters

We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filters are beyond ZFC-inconsistency.     

Models of : when two elements are necessarily order automorphic

We are interested in the question of how much the order of a non‐standard model of can determine the model. In particular, for a model M, we want to characterize the complete types of non‐standard elements such that the linear orders and are necessarily isomorphic. It is proved that this set includes the complete types such that if the pair realizes it (in M) then there is an element c such that for all standard n, , , , and . We prove that this is optimal, because if holds, then there is M of cardinality ?₁ for which we get equality. We also deal with how much the order in a model of may determine the addition.