Helly's property for $n$-cliques and the degree of a graph

The following main result is proved. Let the maximal clique of a graph $G$ have $n$ vertices, and let the degree of any vertex of $G$ be less than $\lceil \frac{5}{3}n\rceil$. Consider a family of pairwise intersecting $n$-cliques. The the intersection of all cliques from that family has more than $n/3$ vertices. It is shown that the result is sharp.