Can groupwise density be much bigger than the non-dominating number?

We prove that g (the groupwise density number) is smaller or equal to b^+ (the successor of the minimal cardinality of a non-dominated subset of omega^omega)

Strong partition relations below the power set: consistency, was Sierpinski right, II?

We continue here [Sh276] but we do not relay on it. The motivation was a conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2-> [omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section 5 we disprove this and give similar negative results. In section 3 we prove the consistency of the conjecture replacing omega_2 by 2^omega, which is quite large, starting with an Erd\H{o}s cardinal. In section 1 we present iteration lemmas which are needed when we replace omega by a larger lambda and in section 4 we generalize a theorem of Halpern and Lauchli replacing omega by a larger lambda

The first almost free Whitehead group

Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardina

A space with only Borel subsets

Miklos Laczkovich asked if there exists a Haussdorff (or even normal) space in which every subset is Borel yet it is not meager. The motivation of the last condition is that under MA_kappa every subspace of the reals of cardinality kappa has the property that all subsets are F_sigma, however Martin's axiom also implies that these subsets are meager. Here we answer Laczkovich' question

On the Arrow property

Let X be a finite set of alternatives. A choice function c is a mapping which assigns to nonempty subsets S of X an element c(S) of S. A rational choice function is one for which there is a linear ordering on the alternatives such that c(S) is the maximal element of S according to that ordering. Arrow's impossibility theorem asserts that under certain natural conditions, if there are at least three alternatives then every non-dictatorial social choice gives rise to a non-rational choice function. Gil Kalai asked if Arrow's theorem can be extended to the case when the individual choices are not rational but rather belong to an arbitrary non-trivial symmetric class of choice functions. The main theorem of this paper gives an affirmative answer in a very general setting

Strongly dependent theories

We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A

On reaping number having countable cofinality

We prove that if the bounding number (d) is bigger than the reaping number (r), then the latter has uncountable cofinality

What majority decisions are possible

The main result is the following: Let X be a finite set and D be a non empty family of choice functions for (X choose 2) closed under permutation of X. Then the following conditions are equivalent: (A) for any choice function c on (X choose 2) we can find a finite set J and c_j in D for j in J such that for any x not= y in X : c{x,y}=y Leftrightarrow |J|/2<| {j in J:c_j{x,y}= y}| (so equality never occurs) (B) for some c in D and x in X we have |{y: c{x,y}=y}| not= (|X|-1)/2 . We then describe what is the closure of a set of choice functions by majority; in fact, there are just two possibilities (in section 3). In section 4 we discuss a generalization

Quite Complete Real Closed fields

We prove that any ordered field can be extended to one for which every decreasing sequence of bounded closed intervals, of any length, has a nonempty intersection; equivalently, there are no Dedekind cuts with equal cofinality from both sides. Here we strengthen the results from the published version

Superatomic Boolean Algebras: maximal rigidity

We prove that for any superatomic Boolean Algebra of cardinality >beth_omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals. Any of those results are essentially best possible