Related papers: PCF and infinite free subsets

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

In this short note we show that if lambda>aleph_1 is regular and lambda is not the successor of a singular cardinal of cofinality aleph_0, and G is a lambda-free abelian group of size lambda, then there is a free group G' subseteq G of size…

There exists a family $\{B_{\alpha}\}_{\alpha<\omega_1}$ of sets of countable ordinals such that o $\max B_{\alpha}=\alpha$, o if $\alpha\in B_{\beta}$ then $B_{\alpha}\subseteq B_{\beta}$, o if $\lambda\leq \alpha$ and $\lambda$ is a limit…

The original theme of the paper is the existence proof of ``there is < eta_alpha : alpha < lambda > which is a (lambda,J)-sequence for < I_i:i<delta >, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and…

We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of…

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that $\kappa,\lambda$ are infinite cardinals such that $\kappa^{+++} \leq \lambda$, $\kappa^{<\kappa}=\kappa$ and $2^{\kappa}= \kappa^+$, and…

We give some existence/nonexistence statements on universal graphs, which under GCH give a necessary and sufficient condition for the existence of a universal graph of size lambda with no K(kappa), namely, if either kappa is finite or…

We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not…

We show a new proof for the fact that when $\kappa$ and $\lambda$ are infinite cardinals satisfying $\lambda ^ \kappa = \lambda$, the cofinality of the set of all functions from $\lambda$ to $\kappa$ ordered by everywhere domination is…

Let $X$ be a set of cardinality $\kappa$ such that $\kappa^\omega=\kappa$. We prove that the linear algebra $\mathbb{R}^X$ (or $\mathbb{C}^X$) contains a free linear algebra with $2^\kappa$ generators. Using this, we prove several…

We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian…

If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…

Let \lambda be a cardinal with \lambda=\lambda^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality \lambda and characteristic p such that the automorphism group of K_0 is a free group of…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

We prove the consistency of the following statement: for some kappa<2^{aleph_0}, there is a kappa-complete ideal on kappa such that the Boolean algebra P(kappa)/I is sigma-centered and there are Q-sets of reals.

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded…

We continue the investigations in the author's book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S_{<= aleph_0}(kappa), subseteq) for kappa real valued measurable (Section 3), densities of box…

Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [lambda]^{<kappa} with no unbounded subsets which are not stationary, where, of course, kappa is regular uncountable less or…

The pcf theorem (of the possible cofinality theory) was proved for reduced products prod_{i< kappa} lambda_i/I, where kappa < min_{i< kappa} lambda_i. Here we prove this theorem under weaker assumptions such as wsat(I)< min_{i< kappa}…

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…