The asymptotic Erd{ő}s--Frankl--Füredi cover-free conjecture

In 1985, Erd{ő}s, Frankl and F{ü}redi conjectured that the threshold $T(r)$ for the existence of an $r$-cover-free family with more members than points satisfies $T(r)=(1+o(1))r^2$. We confirm this conjecture by proving $T(r)\ge r^2+r+2$ for every $r\ge 1$.