Related papers: Open and solved problems concerning polarized part…

We prove the consistency of a strong polarized relation for a cardinal and its successor, using pcf and forcing

We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section…

We force $2^\lambda$ to be large and for many pairs in the interval $(\lambda,2^\lambda)$ a stronger version of the polarized partition relations hold. We apply this toproblem in general topology

We prove the consistency of $\binom{\mu^+}{\mu}\nrightarrow\binom{\mu^+ \omega_1}{\mu\ \mu}$ where $\mu$ is a strong limit singular cardinal of countable cofinality. This result can be forced at limit of measurable cardinals and at small…

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 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…

The main result is that for lambda strong limit singular failing the continuum hypothesis (i.e. 2^lambda > lambda^+), a polarized partition theorem holds.

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^{\mu} > \mu^+$ then $\binom{\mu^+}{\mu}\to \binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every ordinal $\tau<\mu^+$. We obtain an optimal positive relation under $2^\mu = \mu^+$,…

This paper considers three different partition relations from partition calculus, two of which are pair relations and one of which is a triple relation. An examination of the first partition relation and the ramification argument used to…

We show how the separability problem is dual to that of decomposing any given matrix into a conic combination of rank-one partial isometries, thus offering a duality approach different to the positive maps characterization problem. Several…

We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal…

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

Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview…

We prove a positive polarized cube relation for infinite cardinals.

A scalar integer partition problem asks for a number of nonnegative integer solutions to a linear Diophantine equation with integer positive coefficients. The manuscript discusses an algorithm of derivation of linear relations involving the…

We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented)…

We argue that the exactly computable, angle dependent, Casimir force between parallel plates with different directions of conductivity can be measured.

A possible method to solve the sign problem is developed by modifying the original theory. Considering several modifications of the partition function, the observable in the original theory is reconstructed from the identity connecting the…

We solve four out of the six open problems concerning critical cardinalities of topological diagonalization properties involving tau-covers, show that the remaining two cardinals are equal, and give a consistency result concerning this…

In this paper we solve the polarization problem for real Hilbert spaces, a long-standing conjecture that had remained open for nearly three decades. We also confirm that the only extremal configurations are orthonormal sets. These are…