Vs.

Let F be a set of n lines in R3, such that no four lines have a common point of intersection, and no three lines lie in a common plane. Show that there is a subset F F of lines such that no three of them have a common point of intersection and with cardinality |F| cn^3/4 for some absolute constant c > 0.

Hint: Define a random subset S of F by retaining each line independently with probability p.