49 research outputs found
Polarized relations at singulars over successors
Erdos, Hajnal and Rado asked whether
and whether
.
We prove that both relations are independent over ZFC. We shall also prove that
is independent over
ZF for some of cofinality .Comment: 17 pages, to appea
Many Normal Measures
We characterize the situation of having many normal measures on a measurable
cardinal. We show the plausibility of having many normal measures on each
compact cardinal.Comment: 12 page
Bipartite graphs and monochromatic squares
We prove that consistently every bipartite graph of size
contains either a clique or an independent subset of
size for every , where is a
successor cardinal
Simple wedge points
Let V be a finite set of points in the plane, not contained in a line. Assume
|V| = n is an odd number, and |L \cap V| \leq 3 for every line L which is
spanned by V. We prove that every simple line L_{a,b} in V creates a simple
wedge (i.e., a triple {a, b, c} \subseteq V such that L_{a,b} and L_{a,c} are
simple lines). We also show that both restrictions on V (namely |V| is odd and
|L \cap V| \leq 3) are needed. We conjecture, further, that if |V | = n is an
odd number then V contains a simple wedge, even if V is not 3-bounded. We
introduce a method for proving this, which gives (in this paper) partial
results.Comment: 10 page
Separating properties for normal ultrafilters
We define separating properties for normal ultrafilters. We prove that
compactness and supercompactness are separable, yet compactness and
measurability are not. We describe how to use separating properties in order to
elicit distinct normal ultrafilters which do not contain the measurables.Comment: 14 page
Weak diamond and Galvin's property
We prove that the Devlin-Shelah weak diamond implies Galvin's property. On
the other hand, Galvin's property is consistent with the negation of the weak
diamond, and even with Martin's axiom. We show that the proper forcing axiom
implies a relative to the negation of Galvin's property for
Dense free sets
We prove the existence of infinite dense free sets (in the usual topology)
for set mappings on the reals, under reasonable assumptions
On DEPTH and DEPTH^+ of Boolean Algebras
We show that the Depth^+ of an ultraproduct of Boolean Algebras, can not jump
over the Depth^+ of every component by more than one cardinality. We can have,
consequently, similar results for the Depth invariant
Magidor cardinals
We define Magidor cardinals as J\'onsson cardinals upon replacing colorings
of finite subsets by colorings of -bounded subsets. Unlike J\'onsson
cardinals which appear at some low level of large cardinals, we prove the
consistency of having quite large cardinals along with the fact that no Magidor
cardinal exists
Depth^+ and Length^+ of Boolean algebras
Let inv denote the cardinal invariants Depth^+ and Length^+ on Boolean
algebras. For many singular cardinals we create a strict inequality between the
product of the inv values and the inv of the product algebra. The proof holds
in ZFC.Comment: 11 pages, accepted version after minor change