165 problems

Let $X\subseteq\mathbb{R}$ be a measure universal set, meaning that every Lebesgue measurable subset of $\mathbb{R}$ with positive Lebesgue measure contains an affine copy of $X$.…

Does ZFC prove that there exists a simple graph $G=(V,E)$ such that $\chi(G)\geq\omega_1$, $\chi(\overline{G})\geq\omega_1$, and $\chi(G\times\overline{G})=\omega$, where…

For every integer $k\ge 2$, if $\mathcal A\subseteq 2^{[2k]}$ is an intersecting temperate family, then … Moreover, equality is conjectured to hold only for a family of the followi…

Let $E$ be a Banach space, and let $(y_n^*)_{n=1}^{\infty}\subset S_{E^*}$ be a normal sequence, meaning that $y_n^*(x)\to 0$ for every $x\in E$. Must $(y_n^*)$ have a subsequence…

$\forall A\,\bigl(A\text{ is an }AW^*\text{-algebra}\Rightarrow A\text{ is normal}\bigr).$

Can there exist a model $V$ of $\mathrm{ZFC}$ in which $\aleph_\omega$ is the least cardinal $\kappa$ such that $\mathcal{P}^V(\kappa)\neq\mathcal{P}^{\mathrm{HOD}^V}(\kappa)$; tha…

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…

For a finite triple system $S$, consider triple systems that omit $S$ and have uncountable chromatic number. (a) If one exists, must one exist with cardinality at most…

Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$.

For every $x\in\mathbb{R}$ let $A_x\subset \mathbb{R}$ be a bounded set with outer measure $<1$. Must there exist an infinite independent set, that is, some infinite…

Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such that $\cup A_i$ can always be…

Determine the least integer n for which no algorithm decides whether an arbitrary integer-coefficient polynomial in at most n variables has an integer zero. Source: William Gasarch…

Determine whether Con(ZFC plus the existence of a strongly compact cardinal) implies Con(ZFC plus the existence of a supercompact cardinal), or prove that the latter has strictly g…

Select arbitrary finite order k=1, rational constant coefficients a1,...,ak and rational initial values u0,...,u(k-1), each finitely encoded in binary, with u(n+k)=sum(i=1)^k ai u(…

Determine the exact large-cardinal consistency strength of ZFC plus the Proper Forcing Axiom (PFA): identify its equiconsistency level by matching upper and lower consistency bound…

Determine whether ZFR is consistent, where ZFR is the first-order theory ZF(j) plus the assertion that a nontrivial function symbol j:V-V is Sigma1-elementary and hence fully eleme…

Assuming delta is an extendible cardinal, prove that there is an inner model N contained in HOD such that N is a weak extender model for the supercompactness of delta and N satisfi…

Let $G$ be a graph with chromatic cardinal $\aleph_1$. Is there an edge-coloring of $G$ using exactly $\aleph_1$ colors such that, for every vertex-coloring using at most countably…

Let $\kappa$ be an uncountable cardinal, i.e. $\aleph_0<\kappa$. Must there exist a cardinal $\mu$ such that every graph $G$ with chromatic cardinal $\mu$ contains a subgraph $H$ t…

Let $r\in\mathbb N$ with $r\ge2$, let $\lambda$ be an infinite cardinal, let $\gamma$ be an ordinal with $\gamma\ge2$, and let $(\kappa_\alpha)_{\alpha<\gamma}$ be a family of card…

For sets $A_1\subseteq A$, $B_1\subseteq B$, and $C_1\subseteq C$, a box $A_1\times B_1\times C_1$ is monochromatic under a coloring $f:A\to B\to C\to\operatorname{Fin}2$ if there…

For each $d≥1$, can $R^d$ be partitioned into countably many sets such that within each set every positive distance occurs for at most one unordered pair of points?

Let $\mathfrak m$ be a cardinal satisfying $\aleph_0<\mathfrak m<\mathfrak c$, where $\mathfrak c=2^{\aleph_0}$. Suppose $F$ is a family of entire functions…

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely vertex-connected?

No. The following assertion is false: every simple graph $G$ whose chromatic cardinal is $\aleph_1$ contains a subgraph $H$ whose chromatic cardinal is $\aleph_1$ and which is infi…