2 problems

Let $[m]^n$ denote the $n$-dimensional grid poset, and let $\bar{s}([m]^n)$ be its average jump number. The parameters $m$ and $n$ range over values for which…

Centered-set conjecture. Given $m$, there exists a number $d_0(m)$ such that if $d\geq d_0(m)$, then the sets minimizing the number of $k$-chains among subsets of $[m]^d$ of any pr…