9 problems

Unique-overlap conjecture. For all integers $k_1,k_2\in [0,d]$, the intersection $S_{k_1}\cap S_{k_2}$ consists of exactly one combinatorial type, namely the type of $C_{d-1}$.

Generic central slice conjecture. The upper bound on the number of vertices of a $k$-dimensional slice of $C_d$ can be attained by a generic central $k$-dimensional slice for every…

Vertex-number gap conjecture. There are no $(d-1)$-dimensional slices of the cube $C_d$ with $2^i$ vertices, for any $i$ satisfying $2^i\leq 2d-3$.

Isoperimetric conjecture on the n-dimensional cube. The conjecture predicts that on $[0,1/2]$,

Isoperimetric conjecture on the three-dimensional cube. For every $\bar v\in(0,1)$, at least one of the following sets or its complement is an isoperimetric minimizer in…

A sufficiently large integer means an integer exceeding some fixed bound. Four-cubes conjecture. It is conjectured that every sufficiently large integer is a sum of four non-negati…

Exponential upper-bound conjecture. It is conjectured that this bound can be improved to

For each integer $d\ge 2$, let $S_d$ be the set of integers $n$ such that the $d$-dimensional unit cube can be decomposed into $n$ smaller cubes, and let $n_0(d)$ be such that ever…

Mean-square width conjecture. The second moment should satisfy