Polarized relations at singulars over successors

Erdos, Hajnal and Rado asked whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_0}_2$ and whether $\binom{\aleph_{\omega_1}}{\aleph_2}\rightarrow\binom{\aleph_{\omega_1}}{\aleph_1}_2$. We prove that both relations are independent over ZFC. We shall also prove that $\binom{\mu}{\aleph_2}\rightarrow\binom{\mu}{\aleph_2}_2$ is independent over ZF for some $\mu$ of cofinality $\omega_1$.Comment: 17 pages, to appea

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

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 $\kappa^+\times\kappa^+$ contains either a clique or an independent subset of size $\tau\times\tau$ for every $\tau\in\kappa^+$, where $\kappa$ is a successor cardinal

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

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

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 $\aleph_1$

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 $\aleph_0$-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