Jonathan Schilhan — Page 2

arXiv2019

Coanalytic Ultrafilter Bases

We study the definability of ultrafilter bases on $ω$ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in $L$ we can construct $Π^1_1$ P-point and Q-point bases. We also show that the existence of a $\mathbfΔ^1_{n+1}$ ultrafilter is equivalent to that of a $\mathbfΠ^1_n$ ultrafilter base, for $n \in ω$. Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.

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arXiv2019

Towers and gaps at uncountable cardinals

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\mathfrak p(κ)=\mathfrak t(κ)$ or there is a $(\mathfrak p(κ),λ)$-gap of club-supported slaloms for some $λ< \mathfrak p(κ)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\mathfrak p=\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\mathfrak p(κ)$ is always regular; the latter extends results of Garti. Finally, we turn to club variants of $\mathfrak p(κ)$ and present a new model for the inequality $\mathfrak{p}(κ) = κ^+ < \mathfrak{p}_{cl}(κ) = 2^κ$. In contrast to earlier arguments by Shelah and Spasojevic, we achieve this by adding $κ$-Cohen reals and then successively diagonalising the club-filter; the latter is shown to preserve a Cohen witness to $\mathfrak{p}(κ) = κ^+$.

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