Large continuum,oracles

Our main theorem is about iterated forcing for making the continuum larger than ℵ₂. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ₁,ℵ₂ by λ, λ ⁺ (starting with λ = λ ^<λ > ℵ₁). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ ⁺ but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.     

Reflexive abelian groups and measurable cardinals and full MAD families

Answering problem (DG) of [1], [2], we show that there is a reflexive group of cardinality equal to the first measurable cardinal.     

On the p-rank of Ext(A,B) for countable abelian groups A and B

In this note we show that the p-rank of Ext(A,B) for countable torsion-free abelian groups A and B is either countable or the size of the continuum.     

Models of Cohen measurability

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.     

The f-factor Problem for Graphs and the Hereditary Property

If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size ℵ₁. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f-factor. We also discuss the ℵ₂-case.This paper was supported by the Volkswagen Stiftung     

More forcing notions imply diamond

We prove that the Sacks forcing collapses the continuum onto , answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses then it forces . Received July 4, 1994     

The essentially free spectrum of a variety

We partially prove a conjecture from [MeSh] which says that the spectrum of almost free, essentially free non-free algebras in a variety is either empty or consists of the class of all successor cardinals. The author was supported by the NSERC. Prof. Mekler died on June 10, 1992. The author is supported by the United States-Israel Binational Science Foundation; publication 417. The author is supported by the Schweizer Nationalfonds.