, 2015

We consider the graph with 22 vertices $0, 1, \ldots, 21$. Two vertices $x$ and $y$ are connected by an edge if $x - y \equiv \pm 3, \pm 9 \pmod{22}$. We consider only $\pm 3$ modulo 22, then we have a cycle through all vertices. If we choose any $n \geq 12$ vertices, there are two vertices (corresponding to 2 people) with exactly 2 chairs between them. If $n = 11$, we can choose 11 vertices: $0, 6, 12, \ldots, 60 \pmod{22}$. It is clear that $10x \not\equiv \pm 3, \pm 9 \pmod{22}$ for every $x$. It means there is an arrangement for 11 people such that no two people with exactly 2 or 8 chairs between them.