Along some geometrical speculations, I came across a graph $\Gamma$ defined as follows:
Let $S$ be the set of vertices of a regular $n$-gon. Then the "vertices" of $\Gamma$ are the nonempty subsets of $S$ of fixed cardinality $k\leq n$. Two vertices $A,B\subset S$ are connected by an edge in $\Gamma$ iff $$Co(A)\cap Co(B)\neq\emptyset$$
where $Co(X)$ is the convex hull of $X$. So, $A$ and $B$ are connected if either they have a nonempty intersection, or they are "linked" in $\mathbb{S}^{1}$. In such a way, we get some graph $\Gamma(n,k)$.
My question: Does this graph have a name? Does it have some interesting properties? It seems pretty symmetric to me, in some sense.
Note that the graph $\Gamma(n,k)$ is "interesting" only for $k\leq\left[ \frac{n}{2}\right] $, otherwise it is a clique.