Open-problem statements, with their sources attached.

Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\gt c \alpha^2$?

Determine which countable ordinals $β$ have the property that, if $α = \omega^β$, then in any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?

Does the 6-sphere admit a complex structure, i.e. an atlas of holomorphically compatible charts relating it to EuclideanSpace ℂ (Fin 3)?

Are $e$ and $\pi$ algebraically independent?

(B) Existence and smoothness of Navier–Stokes solutions in ℝ³/ℤ³.

The MLC conjecture, stating that the mandelbrot set is locally connected.

The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\infty$ converges. (Note that we 0-index the series below.)

Problem 4.1.* Let $\Omega \subset \mathbb{R}$ be a finite union of intervals and $\nu$ a weak tiling measure for $\Omega$. Must $\mathrm{supp}(\nu)$ have bounded density?

Let $K \subset \mathbb{R}^n$ be a balanced compact set (that is, $\lambda K \subseteq K$ whenever $|\lambda| \leq 1$) and suppose that the normalised Gaussian measure $\gamma_n(K) \geq 0.99$. Does $10K$ contain a compact convex set $C$ with $\gamma_n(C) \geq 0.01$?

Conjecture 1.6 (Even case).* For a nonempty isolate-free graph $G$ on $n$ vertices, if $D$ is even, then $(D + 2)^2 \cdot i(G) \leq (D^2 + 4) \cdot n$.

Is there some $c > 0$ such that every measurable $A \subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?

Stronger conjecture: diameter $\geq n - 1$ for sufficiently large $n$.

It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$. See Conjecture 1.17 in [Wang2017].

NP ≠ coNP*:

The conjecture that the complexity classes NP and coNP are not equal.

The negation of Finite.Equation677_not_implies_Equation255.

Probably this is false.

Conjecture 1.1*: For any odd prime $k$, the sum associated with the classical theta function $θ_3$, $S(k)$ is positive.

Probably there is no such $A$ for the polynomial $X^3$.

The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function f : ℝⁿ → ℝ, there exists a finite set of polynomials gᵢⱼ ∈ ℝ[x₁, ..., xₙ] such that f = supᵢ infⱼ(gᵢⱼ).

Benchmark open subproblem: existence of a SIC-POVM in dimension $59$.