Related papers: More on Halfway New Cardinal Characteristics
We investigate some variants of the splitting, reaping, and independence numbers defined using asymptotic density. Specifically, we give a proof of Con($\mathfrak{i}<\mathfrak{s}_{1/2}$), Con($\mathfrak{r}_{1/2}<\mathfrak{b}$) and…
We investigate several relations between cardinal characteristics of the continuum related with the asymptotic density of the natural numbers and some known cardinal invariants. Specifically, we study the cardinals of the form…
We prove the consistency of $\mathrm{add}(\mathcal{N})<\mathrm{cov}(\mathcal{N})<\mathfrak{p}=\mathfrak{g}=\mathfrak{s}<\mathrm{add}(\mathcal{M})=\mathrm{cof}(\mathcal{M})<\mathfrak{a}=\mathrm{non}(\mathcal{N})=\mathfrak{c}$ with ZFC where…
Based on the well-known cardinal characteristics $\mathfrak{s}$, $\mathfrak{r}$ and $\mathfrak{i}$, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection…
It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four…
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.
We study the consistency and consistency strength of various configurations concerning the cardinal characteristics $\mathfrak{s}_\theta,\mathfrak{p}_\theta,\mathfrak{g}_\theta,\mathfrak{r}_\theta,\mathfrak{t}_\theta$ at uncountable regular…
Assuming three strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) <…
We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different,…
In this article we investigate the dual-shattering cardinal H, the dual-splitting cardinal S and the dual-reaping cardinal R, which are dualizations of the well-known cardinals h (the shattering cardinal, also known as the distributivity…
In [CMRM24], it was proved that it is relatively consistent that \emph{bounding number} $\mathfrak{b}$ is smaller than the uniformity of $\mathcal{MA}$, where $\mathcal{MA}$ denotes the ideal of the meager-additive sets of $2^{\omega}$. To…
We introduce and analyze a new cardinal characteristic of the continuum, the \emph{splitting number of the reals}, denoted $\mathfrak{s}(\mathbb R)$. This number is connected to Efimov's problem, which asks whether every infinite compact…
The paper establishes several inequalities between cardinal characteristics of the continuum. In particular, it is shown that the partition splitting number is not larger than the uniformity of the meagre ideal; not all sets of reals having…
The consistency of a second-order version of a theorem of Morley on the number of countable models was proved in arXiv:2107.07636 with the aid of large cardinals. We here dispense with them.
We give a survey of cardinal charcteristics of the higher Cicho\'n diagram defined on the higher Baire space ${}^\kappa\kappa$ for $\kappa$ regular with $2^{<\kappa}=\kappa$. Specifically, we will compare consistency proofs from the…
The \emph{linear refinement number} $\mathfrak{lr}$ is the minimal cardinality of a centered family in $[\omega]^\omega$ such that no linearly ordered set in $([\omega]^\omega,\subseteq^*)$ refines this family. The \emph{linear excluded…
We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncountable regular cardinal $\kappa$,…
The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the…
We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}\kappa$-sequences (for some regular $\kappa$). As an application, we show that consistently the following cardinal characteristics…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…