Simplicial Complexes with Many Facets are Vertex Decomposable

Abstract

Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$, then $\Delta$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $\Delta$ is vertex decomposable. We give examples to show that this bound is optimal, and that the conclusion cannot be strengthened to the class of matroids or shifted complexes. We explore an application to Simon's Conjecture and discuss connections to other results from the literature.