127 problems

For every pair of linear orders $A$ and $B$ and every $n,m,k,l\in\mathbb{N}_{\geq 1}$, if the ordered sums satisfy $nA+mB\cong kB+lA$, where $nA$ denotes the ordered sum of $n$ cop…

Shelah's three cardinal and model-existence claims. The following assertions hold:

Existence conjecture. There are Shelah–Steprāns almost disjoint families in ${\mathbf{V}}[{G}_{{\omega}_{2}}]$, necessarily of size ${\aleph}_{2}$.

Characterizability conjecture.

Let $\lambda$ and $\kappa$ be cardinals, with $\lambda=\lambda^{<\lambda}>\kappa^+$, $\kappa=\operatorname{cf}(\kappa)$, and $\lambda\ne\aleph_1$. The weaker alternative allows…

Let $\mu$ be a limit of limit cardinals, and let $\lambda<\mu$ range over arbitrarily large regular cardinals. Write $\forall^*$ for “for every large enough.” Shelah's pcf conjectu…

Let ${\mathfrak a}$ be a set of regular cardinals such that every member of ${\mathfrak a}$ is greater than $|{\mathfrak a}|$. Shelah's pcf boundedness conjecture. For no inaccessi…

Let $IFS$ be the axiom schema asserting that for every weakly absolutely definable set $X$ and every partial order $P$ definable in $L(X)$, if … then there is an $L(X)$-generic sub…

Let $\kappa$ be an infinite cardinal. In the game $SMG([\kappa]^{<\aleph_{0}})$, player ONE plays an increasing sequence of members of the generated ideal and TWO responds with mem…

Telgarsky's conjecture. For each positive integer $k$ there is a topological space $(X_k,\tau_k)$ such that TWO does not have a winning $k$-tactic, but does have a winning $k+1$-ta…

A reflexive abelian group is an abelian group naturally isomorphic to its double dual. Existence conjecture. There are reflexive abelian groups of arbitrarily large cardinalities.…

Let $\lambda$ be a singular cardinal, let $\mu<\lambda$, and let $D$ be an ultrafilter that is … -complete and $(\mu,\kappa)$-regular for every $\kappa<\lambda$. Regularity limit p…

Let $X$ be a set of reals of cardinality $\aleph_1$, let $\omega_1$ be the first uncountable ordinal, let $\omega_1^*$ be its converse order, and let $C$ be a Countryman line, mean…

A basis of topological spaces means that every space in the class has one of the listed spaces as the relevant basis representative. Gruenhage's three-element basis conjecture. The…

For a coloring of unordered pairs of $\omega_1$ with two colors, write $\omega_1\rightarrow(\omega_1;\omega_1)^2_2$ for the stated polarized partition relation. Galvin's partition…

Let $X$ be a regular Hausdorff space. A space is hereditarily separable if every subspace is separable, and hereditarily Lindelöf if every subspace is Lindelöf. Hajnal–Juhász basis…

Let $R$ be a binary relation. Write $\fulldiamond$ for the proper forcing axiom appearing in the source, and let $\beta\to(\beta_1;\beta_2)^2_2$ denote the corresponding partition…

Consistency-strength claim. The consistency strength of the theory $

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 $X$ be a set of reals with strong measure zero, and suppose that $|X|<\mathfrak{b}$, where $\mathfrak{b}$ is the bounding number. A finite power of $X$ is a Cartesian product…

A MAD family is a maximal almost disjoint family of infinite subsets of $\text{omega}$. A family is concentrated on a countable subset of itself if there is a countable subfamily s…

A MAD family is a maximal almost disjoint family of infinite subsets of $\text{omega}$, and a subset $X\subseteq 2^\omega$ is a sigma-set if for every Borel set…

Consistency conjecture. It is consistent with ZF plus the Axiom of Dependent Choices (DC) that BPI holds and there is no Vitali set in the real line.

Mathias's question. In Solovay's model, are there infinite MAD families? More generally, is it a theorem of $\mathsf{ZF}+\mathsf{DC}_{\mathbb R}$ that there are no infinite MAD fam…

Let $\varphi$ be an untranscendable linear order. A linear order is strongly indecomposable when [the source context does not provide a definition]. An uncountable real type is the…