A non-face characterization of spheres on few vertices

We prove a relatively simple combinatorial characterization of simplicial $d$-spheres on $d+4$ vertices. Our criteria are given in terms of the intersection patterns of a simplicial complex’s family of minimal non-faces. Namely, let $\mathrm{\Sigma }$ be a simplicial complex on $d+4$ vertices and let $\mathcal{F}$ be its family of minimal non-faces. Then $\mathrm{\Sigma }$ is a $d$-sphere if and only if $\mathrm{\mid }\mathcal{F}\mathrm{\mid }=n\ge 3$ is odd and there is an ordering $A_0,\dots ,A_{n-1}$ of the minimal non-faces, indices taken modulo $n$, such that successive $A_i$ are disjoint and the alternating $\frac{(n-1)}{2}$-fold intersections $A_i\cap A_{i+2}\cap A_{i+4}\cap \cdots \cap A_{i+n-3}$ partition the vertex set.