PCF and infinite free subsets

TL;DR

This paper explores conditions under which large cardinal properties imply the existence of free sets and independent subsets, with applications to Boolean algebra constructions.

Contribution

It provides new proofs relating pcf theory to free sets, and investigates the implications of independence conditions on cofinalities and algebraic structures.

Findings

Large pcf(a) implies existence of free sets.

If pp(aleph_omega)> aleph_{omega_1}, then certain algebraic independence properties hold.

Results extend to non-stationary ideals and Boolean algebra applications.

Abstract

We give another proof that for every lambda >= beth_omega for every large enough regular kappa < beth_omega we have lambda^{[kappa]}= lambda, dealing with sufficient conditions for replacing beth_omega by aleph_omega. In section 2 we show that large pcf(a) implies existence of free sets. An example is that if pp(aleph_omega)> aleph_{omega_1} then for every algebra M of cardinality aleph_omega with countably many functions, for some a_n in M (for n< omega) we have a_n notin cl_M({a_l: l not= n, l<omega}). Then we present results complementary to those of section 2 (but not close enough): if IND(mu,sigma) (in every algebra with universe lambda and <= sigma functions there is an infinite independent subset) then for no distinct regular lambda_i in Reg backslash mu^+ (for i< kappa) does prod_{i< kappa} lambda_i/[kappa]^{<= sigma} have true cofinality. We look at IND(< J^{bd}_{kappa_n}:n<omega >) and more general version, and from assumptions as in section 2 get results even for the non stationary ideal. Lastly, we deal with some other measurements of [lambda]^{>= theta} and give an application by a construction of a Boolean Algebra.