  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.
