Related papers: Hungarian Cubes
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 list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $\mu$ one can force the strong polarized relation with respect to the pair $\mu^+,\mu$.
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 a strong polarized relation for a cardinal and its successor, using pcf and forcing
We introduce "neutrabelian algebras", and prove that finite, hereditarily neutrabelian algebras with a cube term are dualizable.
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 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 show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.
We proved that there are infinitely many cousin primes.
We prove the positivity conjecture for all skew-symmetric cluster algebras.
We mainly show a supercongruence for a truncated series with cubes of Catalan numbers which extends a result by Zhi-Wei Sun.
We investigate negative square-brackets partition relations at successors of singular cardinals of countable cofinality. Along the way we prove some club-guessing results.
We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.
We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations…
In this paper we prove that all irrational numbers from totally real cubic number fields are well approximable by rationals (i.e. the partial quotients in the continued fraction expansion of such a number are unbounded). This settles the…
In this note, we prove that there exist infinite dimensional excellent rings.
A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Its existence is equivalent to the existence of a perfect cuboid with all…
We prove that the non-vanishing conjecture holds for generalized lc pairs with a polarization.
We show that infinitely many cubic fields have class group of 2-rank 1.
We consider rational surfaces $Z$ defined by divisorial valuations $\nu$ of Hirzebruch surfaces. We introduce the concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and…