KöMaL 2009 A494

Let $p_1$,...,$p_k$ be prime numbers, and let $S$ be the set of those integers whose all prime divisors are among $p_1$,...,$p_k$. For a finite subset $A$ of the integers let us denote by $\mathcal{G}(A)$ the graph whose vertices are the elements of $A$, and the edges are those pairs $a$,$b \in A$ for which $a$-$b \in S$. Does there exist for all $m \ge$3 an $m$-element subset $A$ of the integers such that ($a$) $\mathcal{G}(A)$ is complete? ($b$) $\mathcal{G}(A)$ is connected, but all vertices have degree at most 2?