Kalai's Abstract Polynomial Hirsch Conjecture

Let ${\mathcal G}'(k,n)$ be the collection of graphs whose vertices are labeled by $k$-subsets of an $n$-element set, with the property that for vertices labeled by $S$ and $T$, there is a path between them all of whose vertex labels contain $S\cap T$. Kalai's Abstract Polynomial Hirsch Conjecture (APHC). If $G\in {\mathcal G}(k,n)$ is connected, then the diameter of $G$ is bounded above by a polynomial in $k$ and $n$. This conjecture abstracts the Hirsch bound for polytope graphs and motivates an algebraic formulation through linear presentations of square-free monomial ideals; its resolution status is not specified in the source.