7 problems

Nim-number spectrum conjecture. The spectrum of the nim number of avoidance games played in convex geometries is all nonnegative integers

Let $\eta$ and $\zeta$ be the combinatorial invariants under discussion, and let $\overline{\eta}$ denote the relevant conjugate or barred invariant. Suppose that $\zeta$ is the co…

A bouquet of convex geometries is the combinatorial structure considered in the paper as a generalization of ideals of convex geometries. Bouquet realizability conjecture. Every bo…

Let a median system be a set system whose 1-inclusion graph is a median graph, and let $d$ denote its dimension. Median-system realization conjecture. Every median system of dimens…

Let $\mathcal{I}=(U,\mathcal{C'})$ be an ideal of a convex geometry, with Euclidean dimension $\operatorname{dim_{\mathbb{E}}}(\mathcal{I})$ and VC-dimension…

Let $(S,\mathcal{K})$ be a convex geometry, where $\mathcal{K}$ is its family of closed sets. A representation result is known for convex geometries containing the empty set, namel…

Morris's conjecture. For convex geometries generated by such planar point sets, one has