The asymptotic Erdős--Frankl--Füredi cover-free conjecture

In 1985, Erdős, Frankl and Füredi 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 $r\ge1$.