SCHWEITZER 2009 Problem 2

Let $p_1,\dots,p_k$ be prime numbers, and let $S$ be the set of those integers whose all prime divisors are among $p_1,\dots,p_k$ . 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 $m\geq 3$ an $m$ -element subset $A$ of the integers such that (i) $\mathcal G(A)$ is complete? (ii) $\mathcal G(A)$ is connected, but all vertices have degree at most 2?