12 problems

Let $T_{L,L}$ be the diagonal transfer-matrix sub-block for a triangular-lattice strip of width $L$ with fully toroidal boundary conditions, and let $b^{(L)}_1$ denote the stated a…

For a zero-temperature triangular-lattice Potts antiferromagnet on toroidal strips, let $q_0(L)$ and $q_2(L)$ be the relevant real crossings, and let $\ell$ denote the number of br…

Let $q_2(L)$ denote the crossing of the two largest eigenvalues in the $\ell=2$ sector for a triangular-lattice strip of width $L$ with fully periodic boundary conditions, and let…

Let $q_0(L)$ denote the crossing for a triangular-lattice strip of width $L$ with fully periodic boundary conditions, and let $q_{\rm B}\simeq3.8196717312$ be the value where Baxte…

Let $q_0(L)$ denote the relevant real crossing for a triangular-lattice strip of width $L$ with fully periodic boundary conditions. Triangular-lattice 3p ± 1 conjecture. For all wi…

Let $S$ be a set of $n$ points in the plane. Let $m_1(S)$ and $m_2(S)$ be the numbers of occurrences of the smallest and second smallest distances in $S$, respectively, and let…

Triangular lattice minimizer conjecture. For every $s>2$, the triangular lattice $\mathsf{A}_2$ is the unique minimizer of $\mathcal{F}_s$ among all two-dimensional lattices of uni…

Hypersingular triangular-lattice expansion conjecture. For $2<s<4$,

Collectively jammed density conjecture. If such a packing is collectively jammed and is not a triangular packing, then its density is at most

Density Gap Conjecture. If $n+1$ is a triangular lattice number but $n$ is not, then the most dense packing of $n$ equal disks in the triangular torus has density

Fejes Toth's conjecture. The triangle packing in the plane, with one packing disk removed, is solid.

McDiarmid–Reed conjecture. Every triangle-free induced subgraph of the triangular lattice is