A non-trivial bound for 3AP-intersecting families

A family $F$ of subsets of $[n]$ is 3AP-intersecting if every two members have intersection containing a non-trivial three-term arithmetic progression. We prove that there is an absolute constant $c>0$ such that any such $F$ has size at most $(\tfrac12 - c)2^n$. This is the first non-trivial progres