25 problems

Weaker nonexistence conjecture. There is no Borel morphism from $\bold R$ to $\bold R_3$.

Nonexistence conjecture. There is no Borel morphism from the old product of two copies of $\bold R$ to $\bold R_3$.

Additivity conjecture.

A $4$-ladder is a lattice of breadth $4$; its cardinality is the cardinality of its underlying set. The continuum hypothesis, denoted by $\mathsf{CH}$, asserts that…

Fréchet $\mathfrak{b}$-scale conjecture. There are Fréchet $\mathfrak{b}$-scales in the model constructed in this section.

Dualization conjecture.

Let $\mathfrak{e}_2^\mathrm{const}$ be the least size of a family of binary sequences not constantly predicted by any predictor, and let $\mathfrak{d}$ denote the dominating number…

Let ${\mathsf{all}}$ be the ambient parameter set, and let $X,X'\subseteq{\mathsf{all}}$ satisfy … Let $Y$ be arbitrary, and write ${\mathfrak{dd}}_{X,Y}$ for the associated densit…

For $r\in(0,1)$, let ${\mathfrak{dd}}_{\{r\},{\mathsf{all}}}$ denote the corresponding density cardinal; the same notation is used when ${\mathsf{all}}$ is replaced by $[0,1]$,…

Conjecture. Ketonen's construction of a p-point from $\mathfrak{d}=\mathfrak{c}$ can be modified to produce a non-rapid almost-rapid p-point.

Let $\kappa$ be the inaccessible cardinal under consideration, and let the parameters $b,h\in{}^{\kappa}\kappa$ determine the cardinal characteristics on the $\mathfrak b$- and…

Let $\kappa$ be the inaccessible cardinal under consideration, and let $b,h\in{}^{\kappa}\kappa$. Consider the cardinal characteristics $\mathfrak d_\kappa^{b}(\leq^*)$,…

Splitting-number conjecture. The cardinal $\mathfrak{s}$ is the first cardinal $\kappa$ for which there exists a non-fin-intersecting almost disjoint family of size $\kappa$; equiv…

Near-coherence conjecture. If $\b5d=\b5u=\b5c$, then there exist uncountably many $\le_{\mathrm{RB}}$-incompatible $P$-points.

The ZFC equality conjecture. In $\mathsf{ZFC}$ it is provable that

Partial-slalom separation conjecture.

Total-slalom conjecture. For total slaloms, the generalized slalom characterization is different from the additivity of $\operatorname{id}(\mathbb Q_\kappa)$.

Consistency conjecture. There exists a model $\bo\mathbf V$ such that

Let {\mspace{2mu}\mid\mspace{-12mu} {\raise0.6em\text{bullet}}} denote the stick principle, and let Theorem be the theorem discussed in the source, whose hypothesis involves th…

Let the constellations of Section 2 be the configurations of cardinal characteristics of the continuum obtained there, and suppose their associated cardinal values do not exceed…

Equality conjecture.

Let $\mathcal{I}$ be a fragmented ideal. The associated Rothberger number $\mathfrak{b}(\mathcal{I})$ is the least size of a family witnessing the corresponding Rothberger property…

Nonexistence conjecture. There is no model in which every ideal generated by an $e4df1$-tower is dense only if it is non-meager; equivalently, assuming $d4f=e4df1$, a meager dense…

Cardinal-characteristic hypothesis conjecture. Similar hypotheses involving combinatorial cardinal characteristics of the continuum should settle the problems mentioned above.

Let $lambda<kappa$ be regular uncountable cardinals such that there exists a simple $P_lambda$-point $\mathcal U$ and a $P_kappa$-point $\mathcal F$. Consistency conjecture. It is…