102 research outputs found
The ultrafilter number for singular cardinals
We prove the consistency of a singular cardinal with small value of
the ultrafilter number , and arbitrarily large value of .Comment: 8 page
The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis
The purpose of this paper is to present some results which suggest that the
Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will
be proved is that a form of simultaneous reflection follows from the Set
Mapping Reflection Principle, a consequence of PFA. While the results fall
short of showing that MRP implies SCH, it will be shown that MRP implies that
if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega =
{a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP
always fails in a generic extension by Prikry forcing.Comment: 7 page
More on Compactness of Chromatic Numbers
We prove that for any regular kappa and mu > kappa below the first fix point
(lambda = aleph_lambda) above kappa, there is a graph with chromatic number >
kappa, and mu^kappa nodes but every subgraph of cardinality < mu has chromatic
number less than or equal to kappa.Comment: 8 page
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