8 problems

$k$-Roudneff's conjecture. For any rank $r\geq 2k+1\geq 3$ oriented matroid $\mathcal{M}$ on $n\geq r+1$ elements,

A polytope is neighborly if every set of at most $\lfloor d/2 \rfloor$ vertices forms a face. For any pair $(n,d)$, a cyclic polytope is a $d$-dimensional polytope with $n$ vertice…

Uniqueness conjecture. For $k=1$, the reorientation class $[C_r(n)]$ is the unique reorientation class having $c_r(n,k)$ $k$-neighborly reorientations.

Let $P$ be a 2-neighborly $d$-polytope, meaning that every two vertices of $P$ form an edge. Write $\operatorname{fct}(P)$ for the number of facets of $P$ and…

Neighborly-polytope conjecture. Among all $d$-dimensional rectilinear drawings of $K_n^d$, the number of crossing pairs of hyperedges is maximized when the vertices are placed in g…

Let $P$ be a $d$-dimensional convex polytope, and let $f_i(P)$ denote the number of its $i$-dimensional faces. A polytope is $k$-neighborly if every subset of $k$ vertices is the v…

A polytope is neighborly if every set of at most half the dimension's vertices forms a face, and a polytope is circumscribable if it has a realization whose facets are tangent to a…

Degree threshold conjecture. The weak Cayley conclusion should hold under the strengthened threshold