On sewing neighbourly polytopes

In 1982, I. Shemer introduced the sewing construction for neighbourly 2m-polytopes. We extend the sewing to simplicial neighbourly d-polytopes via a verification that is not dependent on the parity of the dimension. We present also descibable classes of 4-polyopes and 5-polytopes generated by the construction

Convex Independence in Permutation Graphs

A set C of vertices of a graph is P_3-convex if every vertex outside C has at most one neighbor in C. The convex hull \sigma(A) of a set A is the smallest P_3-convex set that contains A. A set M is convexly independent if for every vertex x \in M, x \notin \sigma(M-x). We show that the maximal number of vertices that a convexly independent set in a permutation graph can have, can be computed in polynomial time

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]

Subpolytopes of cyclic polytopes

AbstractA remarkable result of Shemer states that the combinatorial structure of a neighbourly 2 m -polytope determines the combinatorial structure of each of its subpolytopes. From this, it follows that every subpolytope of a cyclic 2 m -polytope is cyclic. In this note, we present a direct proof of this consequence that also yields that certain subpolytopes of a cyclic (2 m+ 1)-polytope are cyclic