3 problems

For a poset $P$, let $h(P,x)$ denote the average height of $x$, and let $\eta(P)$ be the smallest absolute difference $|h(P,x)-h(P,y)|$ over incomparable elements $x,y$. Set … Saks…

TVK-shape sorting-probability conjecture. There is a universal constant $C>0$ such that

The one-third–two-thirds conjecture. For every finite poset $P=(X,\prec)$ that is not a chain, we have