8 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…

Let $P=(X,\prec)$ be a finite poset that is not a chain, and let $f$ be uniformly random in the set $\operatorname{\mathcal E}(P)$ of linear extensions. One-third–two-thirds conjec…

Little-o conjecture. As $n\to\infty$,

Exponent conjecture. The exponents satisfy

One-third–two-thirds conjecture. For every finite poset $P$,

Three-dimensional box sorting-probability conjecture. As $m\to\infty$,

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