Related papers: A comment on "p<t"

We prove that the strong polarized relation for the continuum holds for $\aleph_0$ and for every supercompact cardinal. We use iteration of Mathias forcing.

We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to…

We prove that the strong polarized relation of $\theta$ above $\omega$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal…

We extend the applications of the techniques used in Arch Math Logic 52:261-278, 2013, to present various examples of consistency results where some cardinal invariants of the continuum take arbitrary regular values with the size of the…

We prove that if there is a real-valued measurable cardinal then the splitting number is $\aleph_1$. Likewise, if the continuum is real-valued measurable then the reaping number equals the continuum.

Assuming an inaccessible cardinal kappa, there is a generic extension in which MA + 2^{aleph_0} = kappa holds and the reals have a Delta^2_1 well-ordering.

We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular…

Characteristic earlier results were of the form CON$(2^{\aleph_0} \to [\lambda]^2_{n, 2})$, with $2^{\aleph_0} $ an ex-large cardinal, in the best case the first weakly Mahlo cardinal. Characteristic new results are CON$((2^{\aleph_0} =…

Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a base of cardinality <= mu, lambda <= mu < 2^lambda, lambda strong limit of cofinality aleph_0, then T has…

A usual dichotomy is that in many cases, reasonably definable sets, satisfy the CH, i.e. if they are uncountable they have cardinality continuum. A strong dichotomy is when: if the cardinality is infinite it is continuum as in [Sh:273]. We…

We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak…

Strong reflection principles with the reflection cardinal $\leq\aleph_1$ or $<2^{\aleph_0}$ imply that the size of the continuum is either $\aleph_1$ or $\aleph_2$ or very large. Thus, the stipulation, that a strong reflection principle…

We prove that if cf(lambda) > aleph_0 and 2^{cf(lambda)}<lambda, then lambda->(lambda,omega+1)^2.

If kappa is strongly compact, lambda > kappa is regular, then (2^{< lambda})^+ --> (lambda+eta)^2_theta holds for eta,theta<kappa.

We find the exact best possible range of those $p > 1$ for which any function which belongs to $A_1(\mathbb{R})$, with $A_1$-constant equal to $c$, must also belong to $L^p$. In this way we provide alternative proofs of the results in [2]…

We prove the consistency of: if B_1, B_2 are Boolean algebra satisfying the c.c.c. and the 2^{aleph_0}-c.c. respectively then B_1 x B_2 satisfies the 2^{aleph_0}-c.c.

The $c_2$ invariant is an arithmetic graph invariant related to quantum field theory. We give a relation modulo $p$ between the $c_2$ invariant at $p$ and the $c_2$ invariant at $p^s$ by proving a relation modulo $p$ between certain…

For g < f in omega^omega we define c(f,g) be the least number of uniform trees with g-splitting needed to cover a uniform tree with f-splitting. We show that we can simultaneously force aleph_1 many different values for different functions…

Motivated by results of Juh\'asz and van Mill in [13], we define the cardinal invariant $wt(X)$, the weak tightness of a topological space $X$, and show that $|X|\leq 2^{L(X)wt(X)\psi(X)}$ for any Hausdorff space $X$ (Theorem 2.8). As…

We prove that the cardinality of transitive quasi-uniformities in a quasi-proximity class is at least $2^{2^{\aleph_0}}$ if there exist at least two transitive quasi-uniformities in the class. The transitive elements of $\pi(\delta)$ are…