FKMO 2016 Problem 6

Let $U$ be a set of $m$ triangles. Prove that there exists a subset $W$ of $U$ which satisfies the following.

(i). The number of triangles in $W$ is at least $0.45m^{\frac{4}{5}}$ (ii) There are no points $A, B, C, D, E, F$ such that triangles $ABC$ , $BCD$ , $CDE$ , $DEF$ , $EFA$ , $FAB$ are all in $W$ .