Graph Coloring · Superchallenge problem

Let $G$ have as its vertices the $\binom{6}{2}=15$ two-element subsets of $\{1,2,3,4,5,6\}$, with two subsets adjacent exactly when they are DISJOINT. The largest family of PAIRWISE-INTERSECTING subsets (an independent set of $G$) has size $5$ (e.g. all $5$ pairs containing the element $1$), and for $6$ elements this maximum is achieved ONLY by such "star" families (all pairs sharing one fixed common element). What is $\chi(G)$?