### Decisive creatures and large continuum

by Kellner and Shelah. [KrSh:872]

J Symbolic Logic, 2009

For f>g in omega^omega let c^{for all}_{f,g} be the
minimal number of uniform trees with g-splitting needed to
for all^infty-cover the uniform tree with
f-splitting. c^{exists}_{f,g} is the dual notion for the
exists^infty-cover.
Assuming CH and given aleph_1 many (sufficiently different) pairs
(f_epsilon,g_epsilon) and cardinals kappa_epsilon such that
kappa_epsilon^{aleph_0}= kappa_epsilon, we construct a
partial order forcing that c^{exists}_{f_epsilon,g_epsilon}=
c^{for all}_{f_epsilon,g_epsilon}= kappa_epsilon .
For this, we introduce a countable support semiproduct of decisive
creatures with bigness and halving. This semiproduct satisfies
fusion, pure decision and continuous reading of names.

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