The upper bound theorem for centrally symmetric simplicial spheres

A centrally symmetric (or cs, for short) simplicial complex is cs-\$i\$-neighborly if every set of \$i\$ of its vertices, no two of which are antipodes, forms a face. Adin and Stanley (unpublished) proved that among all cs simplicial spheres of dimension \$d-1\$ and with \$2n\$ vertices, a cs-\$\left\lfloor d/2\right\rfloor\$-neighborly sphere simultaneously maximizes all the face numbers, assuming such a sphere exists.

The existence of cs-2-neighborly simplicial 3-spheres with \$2n\$ vertices for any \$n\geq 4\$ was confirmed by Jockusch in 1995. In this talk I will discuss how to generalize his construction and show that for all \$d \geq 4\$ and \$n\geq d\$, there exists a cs simplicial \$(d-1)\$-sphere with \$2n\$ vertices that is cs-\$\left\lfloor d/2\right\rfloor\$-neighborly. This completely resolves the upper bound problem for cs simplicial spheres.

Joint work with Isabella Novik.