CN-TST 2015 Problem 17

Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$ , and a pair of vertices $(u,v)$ , define $$L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}$$Prove that, for any choice of $V$ , there exist at least $120$ distinct values of $L(u,v)$ .