Related papers: The ultrafilter and almost disjointness numbers

We study the boundary weighted regularity of weak solutions $u$ to a $s$-fractional $p$-Laplacian equation in a bounded smooth domain $\Omega$ with bounded reaction and nonlocal Dirichlet type boundary condition, in the singular case…

We prove that the statement `For all Borel ideals I and J on $\omega$, every isomorphism between Boolean algebras $P(\omega)/I$ and $P(\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every…

A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of Box_lambda for singular lambda is proved. CECA is used to show that certain ``almost point- < tau'' families can be refined to point- < tau…

We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(\omega)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost…

We prove the consistency of $\mathfrak{r}_\lambda<\mathfrak{d}_\lambda$ and even $\mathfrak{u}_\lambda<\mathfrak{d}_\lambda$ for a singular cardinal $\lambda$.

We show that Shelah cardinals are preserved under the canonical $GCH$ forcing notion. We also show that if $GCH$ holds and $F:REG\rightarrow CARD$ is an Easton function which satisfies some weak properties, then there exists a cofinality…

Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable.…

We prove from the existence of a Mahlo cardinal the consistency of the statement that $2^\omega = \omega_3$ holds and every stationary subset of $\omega_2 \cap \mathrm{cof}(\omega)$ reflects to an ordinal less than $\omega_2$ with…

A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erd\H{o}s-Ko-Rado theorem of 1961 that…

The Polyline Bundle Simplification (PBS) problem is a generalization of the classical polyline simplification problem. Given a set of polylines, which may share line segments and points, PBS asks for the smallest consistent simplification…

We prove the existence of infinitely many classical periodic solutions for a class of degenerate semilinear wave equations: \[ u_{tt}-u_{xx}+|u|^{s-1}u=f(x,t), \] for all $s>1$. In particular we prove the existence of infinitely many…

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler's order has continuum many classes, however, the construction does not…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

We find a polynomial in three variables whose values at nonnegative integers satisfy the Erd\H{o}s-Straus Conjecture. Although the perfect squares are not covered by these values, it allows us to prove that there are arbitrarily long…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D \geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…

Let $\Omega \subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in…

This is a follow up to a paper by the author where the disjointness relation for (the graphs of) definable functions from ${^\omega \omega}$ to ${^\omega \omega}$ is analyzed. In that paper, for each $a \in {^\omega \omega}$ we defined a…

We answer an old question of Michael Hru\v{s}\'{a}k by constructing a $+$-Ramsey \textsf{MAD} family without the need of any additional axioms beyond $\mathsf{ZFC.}$ We also prove that every Miller-indestructible \textsf{MAD }family is…

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schr\"odinger equation consists of several equations, one for each time variable. This…