An infinite family of nearly neighborly centrally symmetric 3-spheres (Q1903014)

A simplicial \(d\)-sphere \(S\) with vertex set \(V\) is called centrally symmetric if it admits a fixed-point-free involution (which pairs up the vertices into pairs of antipodal vertices). A centrally symmetric simplicial \(d\)-sphere \(S\) is said to be nearly neighbourly if each subset of \(V\) of cardinality at most \({d + 1 \over 2}\) which does not contain a pair of antipodal vertices, is a face of \(S\). For each \(n \geq 4\), the paper constructs a nearly neighbourly centrally symmetric 3-sphere with \(2n\) vertices. This is of interest for an upper bound theorem for the number of faces of centrally symmetric 3-spheres.