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).