15 problems

Let ${\mathcal D}$ be an alternating knot diagram with no nugatory crossings. The determinant of ${\mathcal D}$ is the order of its first homology group of the double branched cove…

Collapse conjecture.

Extremality conjecture. For all $t,r\geq 2$,

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 $\mathrm{HJ}(t,r)$ be the Hales–Jewett number. The diagonal is…

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.

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 $n\geq12$,

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…

The multiplicative decomposition conjecture. If $a$, $b$, and $n$ are integers, $n>2$, $a>1$, $|b|>1$, and $\gcd(a,b)=1$, then

The uniqueness conjecture. For $n>2$, $c_{2,n}$ is the only $n$-coloring, up to isomorphism, of the nonzero rational numbers without a monochromatic solution to $E(2,n)$.