Cyclic polytopes whose boundary is a flag complex

Question

A cyclic polytope $C(n, d)$ is defined as the convex hull of $n$ distinct points on the moment curve in $\mathbb{R}^d$ (here $n>d$). This is a simplicial polytope so its boundary $\partial C(n, d)$ is a simplicial complex. I would like to know for what values of $n$ the boundary is a flag complex (i.e., a simplicial complex completely determined by its $1$-skeleton).

Answer 1

The answer is "never" (except in the obvious case $d=2$, $n\ge 4$).

$C(n,d)$ is neighborly, meaning that every $d/2$ or less vertices define a simplex. In particular, for $d\ge 4$ its graph is complete and the boundary complex cannot be flag. For $d=3$, the boundary complex contains the three edges $1i$, $in$ and $1n$ but not the triangle $1in$, for every $i\in \{3,...,n-2\}$.