14 problems

Let $[m]=\{1,\ldots,m\}$. For integers $m,n\geq 1$ and tuples $\lambda^{(i)}\in[m]^n$ for $i=1,\ldots,m$, let $\operatorname{\mathcal{T}}_{m,n}(\lambda^{(1)},\ldots,\lambda^{(m)})$…

Explicit asymptotic formula conjecture. If $d$ is odd and

Linear-growth conjecture. For all fixed $d$ and $k$, there exists a constant $C_{d,k}$ such that

Three-dimensional covering conjecture. For all $n,k\geq 1$,

Two-dimensional asymptotic conjecture. For all positive integers $n$ and $k$,

Fractional lower-bound conjecture. For every fixed $k$, the bound $kf^*(n,2)$ is not best possible for all sufficiently large $n$; equivalently, the integer covering number satisfi…

Let $T_{n,k}$ denote the triangular grid obtained after $k$ transformations from the $n$-grid, and let $T_{n,n-s}^s$ denote an $s$-subgrid of $T_{n,n-s}$. For an edge, its edge val…

Let $T_n$ be the triangular grid graph shown in Figure, with $n$ rows and $m=n^2$ cells. Let $a$ and $b$ be distinct vertices of degree $2$, and let $r_n(a,b)$ be the resistance di…

Signed triangular-grid chromatic-number conjecture. The chromatic number of every signed triangular grid is $6$.

Let $P_n$ be the triangular grid graph with $n$ vertices on each side. Let $T_n$ be the triangular number used in the source, and write $\alpha\approx\beta$ when…

Let $P_n$ be the triangular grid graph with $n$ vertices on each side, and let $\partial C$ denote the boundary vertices of a vertex set $C$. A row packing fills each row completel…

Let $P_n$ be an equilateral triangular grid graph with $n$ vertices on each side. Lions clear vertices they visit while contamination spreads along unblocked edges. Triangular-grid…

Consider a strip discretized by a triangular grid graph, with height $n$, and suppose all lions must move simultaneously at every time step. Simultaneous-motion insufficiency conje…

Let $T_n$ be the triangular grid graph with starting vertex $v^0_0$, and consider the feedback game on $T_n$ in which Alice and Bob play according to the game's rules. Alice's winn…