13 problems

Finite-power strong measure zero conjecture. If $X$ has strong measure zero and $|X|<\mathfrak{b}$, then all finite powers of $X$ have strong measure zero.

Let $\lambda$ be a cardinal with $\bf{\delta^1_4}\leq\lambda\leq\bf{\delta^1_6}$, and suppose $\operatorname{cof}(\lambda)>\omega$. A cardinal is $\bf{\Gamma}$-unreachable when the…

Let $\bf{\Gamma}$ be a pointclass, and let $\kappa$ be $\bf{\Gamma}$-reachable when there is a sequence of distinct $\bf{\Gamma}$-sets of length $\kappa$; otherwise, $\kappa$ is…

Let $G$ be a compact topological group. A subgroup $H\leq G$ is dense if its closure in $G$ is $G$, and $d(H)$ denotes the density character of $H$, while $w(G)$ denotes the weight…

Invariance conjecture. For every compact Polish space $X$ with

Tzaban–Paolini–Shelah conjecture. For every infinite metrizable profinite group $G$,

Generalized Borel conjecture. The cardinality of $X$ is no larger than the weight of $G$:

Let $F$ be a countable field, and let $\diamondsuit(\omega^\omega,=^\infty)$ denote the parameterized diamond principle appearing in the source. Diamond principle conjecture. If…

Let $F$ be a countable field, let $\mathrm{non}(\mathcal{M})$ denote the uniformity of the meager ideal, and let $\mathfrak{a}{\mathrm{vec},F}$ be the least cardinality of a mad famil…

Let $(X,\tau)$ be a topological space, and let $PR(X)=[X]^{<\omega}$ be the set of finite subsets of $X$ equipped with the Pixley–Roy topology, whose basic open sets are … for…

Characterization and cardinal-invariant conjecture. begin{enumerate} item A compact space $X$ is D-separable if and only if $X$ has a $\sigma$-disjoint $\pi$-base. item For every c…

Sacks-indestructibility conjecture. If

Nonexistence conjecture. There is a model of