28 problems

Collapse conjecture.

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}(t,r)$ be the Hales–Jewett number. The diagonal is…

Let a finite set $X\subseteq\mathbb{R}^m$ be $2$-Ramsey if every $2$-coloring of $\mathbb{R}^m$ contains a monochromatic congruent copy of $X$. A triangle is non-equilateral if its…

For an integer $r>1$ and a knot $K$, let $\mathrm{mincol}_r K$ denote the minimum number of colors required for a non-trivial $r$-coloring, minimized over all diagrams of $K$. For…

Deng–Tidor–Zhao's conjecture. The interval $[N]$ admits a coloring with $N^{o(1)}$ colors and no nontrivial symmetrically colored 4-term arithmetic progressions.

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}(t,r)$ be the Hales–Jewett number. Let…

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}^{[k]}(t,r)$ denote the least dimension forcing a m…

Periodic coloring conjecture. For any $V$, $N$, $\Sigma$, and $\Omega$ as above, the solution set $\operatorname{Coloring}(V,N,\Sigma,\Omega)$ is not aperiodic.

Optimal-coloring conjecture. The coloring that colors multiples of $4$ red and all other integers blue is optimal for the equation $4x+4y=z$.

Costello–Elvin conjecture. If $\mathcal{E}$ is a three-variable equation of the form $ax+by=cz$ with $a,b,c\in\mathbb{N}$, then

Let $L$ be a prime link with reduced alternating diagram and determinant $b\deltad$. A Fox $b\deltad$-coloring assigns colors to the arcs of the diagram. Generalized Kauffman–Harar…

Conlon–Wu conjecture. For every non-spherical set $X$, there exists a natural number $m$ such that

Erdős et al.'s conjecture. For every non-equilateral three-point configuration $T$, every 2-coloring of $\mathbb{E}^2$ contains a monochromatic congruent copy of $T$ in one of the…

Let ${\cal T}$ be a sphere triangulation with degree sequence $6,\ldots,6,2,2,2$. For a coloring ${\cal C}$ of ${\cal T}$, let $\eta({\cal C})=f^2/F$, where $f$ is its fold count a…

For $s,k\in\mathbb N$, let $A_{s,k}=\{s,2s,\ldots,ks\}$. A coloring is rainbow on a set when all elements of that set receive distinct colors. Rainbow-coloring conjecture. For ever…

Let $\mathcal R^3$ be the unit distance graph of $\mathbb R^3$, and let $\rho_\lozenge=[uvwx]\subset\mathbb R^3$ be the unit rhombus defined in the source. A coloring is a map…

The optimal block-coloring conjecture. For $n\geq12$,

Let $\kappa$ be a cardinal, and let $\operatorname{U}(\kappa,2,\omega,2)$ denote the relevant coloring principle at the parameter $\omega$. Weak compactness characterization at ome…

Let $ba$ be a cardinal, let $b8\requirestext{AMScd}\in\operatorname{Reg}(ba)$, and let $\operatorname{U}(ba,2,b8,2)$ denote the relevant coloring principle. Weak compactness charac…

Consider the discrete cube $\{0,1,2,\ldots,m\}^n$ with the topology of the standard lattice graph, colored with $k$ colors. A monochromatic connected component is a connected compo…

Let $S$ be a finite point set in $\mathbb{R}^d$; a $k$-coloring assigns one of $k$ colors to each point, and a simplex is empty if its interior contains no point of $S$. The many-c…

Let $\mathbb{E}^2$ denote the Euclidean plane, and let a 2-coloring assign one of two colors to every point of $\mathbb{E}^2$. A monochromatic equilateral triangle has all three ve…

Partial coloring extension conjecture. Then $y$ can be extended to a $k$-coloring if and only if $\overline{[y]} \cap k^G$ consists of aperiodic points.

Hyper aperiodic extension conjecture. Then $y$ can be extended to a hyper aperiodic point in $k^G$ if and only if $\overline{[y_*]} \cap k^G$ consists of aperiodic points.

The finite-or-continuum conjecture. For every nonregular finite system $\mathcal{L}$ of linear homogeneous equations, either $\Delta(\mathcal{L};\mathbb{Q})$ is finite or…