Open-problem statements, with their sources attached.

Fuglede's conjecture* in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.

A generalisation of the problem to sets $A \subseteq (0, N]$ of real numbers, such that the subset sums all differ by at least $1$ is proposed in [Er73] and [ErGr80].

[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Estimate an upper bound for$\alpha(n)$.

Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?.

Problem 15 in [Ar2013]: Is every homogeneous ω-monolithic compact hausdorff space first countable?

Conjecture 4.4*: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$), $nk < S(k)$.

Is there any polynomial $f(x, y) \in \mathbb{Q}[x, y]$ such that $f : \mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{Q}$ is a bijection?

Benchmark open subproblem: existence of a SIC-POVM in dimension $69$.

The Köthe conjecture: for any positive integer n, the Köthe radical of R is the matrix ideal M_2(Nil*(R)).

Markel's $S_3$-conjecture* (1973): any nontrivial finite ah-group is isomorphic to $S_3$.

The conjecture is open in general; it is known to be true for solvable groups.

Variant of the Bézier-Bernstein Voronovskaja problem which treats "sufficiently smooth" as an eventual condition in the smoothness order $m$: for all sufficiently large finite $m$, every $C^m$ function on $[0,1]$ should have the asserted asymptotic formula.

First open case beyond Erdős–Rado*: $\mathfrak{c} \to (\omega \cdot 2, 4)^3_2$.

Erdős and Rado proved $\mathfrak{c} \to (\omega + n, 4)^3_2$ for every finite $n \ge 2$ (see erdos_rado), which covers all red ordinals below $\omega \cdot 2 = \omega + \omega$. This variant asks whether the result extends to $\beta = \omega \cdot 2$, the simplest countable ordinal not covered by their theorem.

In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value of the Landau constant.

Is $\pi ^ \pi$ irrational?

De Giorgi's conjecture holds in dimension $n = 5$.

The boundary of any Multibrot set is conjectured to have zero area. Note that we don't need to exclude the trivial cases n = 0 and n = 1 because the conjecture holds for them.

The Kakeya set conjecture: Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$.

Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$?

How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$?

In particular, can one find a better bound than the best-known bound from [KrLe25]?

Is there a dense subset of ℝ^2 such that all pairwise distances are rational?