Open-problem statements, with their sources attached.

Problem 16 in [Ar2013]: Is the cardinality of every homogeneous ω-monolithic compact hausdorff space not greater than 𝔠?

Let D be the diagonal group of SL_n(ℝ) where n ≥ 3. Then any relatively compact D-orbit in SL_n(ℝ) / SL_n(ℤ) is closed.

Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients. If $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ are also $0, 1$ polynomials.

Benchmark open subproblem: existence of a SIC-POVM in dimension $70$.

Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$, $g^n = 1$. Is $G$ necessarily finite?

Existence-only version of the eventual-smoothness variant. This separates the first part of the source problem, proving that the scaled sequence has some limit, from the stronger task of finding an explicit expression for that limit.

*The relation at $\omega_1$**: $\mathfrak{c} \to (\omega_1, n)^3_2$ for finite $n \ge 2$, where $\omega_1 = \aleph_1$ is the first uncountable ordinal.

Note that $\omega_1$ is not a countable ordinal, so this is not directly an instance of the main Erdős problem (which asks for countable $\beta$). Under CH, $\omega_1 = \mathfrak{c}.\mathrm{ord}$, making this a self-referential question about $\mathfrak{c}.\mathrm{ord} \to (\mathfrak{c}.\mathrm{ord}, n)^3_2$.

Brennan's conjecture, part 1: $B(-2) = 1$.

Is $\ln(\pi)$ irrational?

De Giorgi's conjecture holds in dimension $n = 6$.

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers.

Ref: Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$

Is there a better bound than the best-known bound from [KrLe25]? This is an existential version of the main problem that does not require providing the bound explicitly.

Let $n \geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?

Problem 17 in [Ar2013]: Is it true that every nonempty ω-monolithic compact hausdorff space contains a point with a first countable neighborhood basis?

Note: Nonempty X is required since the conclusion asserts the existence of a point.

Problem 10.1. Are there a transcendental number $\alpha$ and a positive real number $\xi$ such that $\lVert \xi \alpha^n \rVert$ tends to~$0$ as~$n$ tends to infinity? [Har19] (Trivial for $|\alpha| < 1$)

The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.

Benchmark open subproblem: existence of a SIC-POVM in dimension $71$.

If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least $e^{\sqrt n}$ in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.

Variant of the Bézier-Bernstein Voronovskaja problem with the required smoothness order itself left as an answer. Replacing (answer(sorry) : ℕ × ((ℝ → ℝ) → ℝ → ℝ)) by a concrete value lets one state the conjecture for a chosen regularity threshold.

Determine the value of the Busy Beaver function at n = 6.