One of the most interesting questions in Mathematics concerns the Mordell-Weil rank of the group of rational points on elliptic curves $E/\mathbb{Q}$, namely whether this quantity is bounded as one varies over all elliptic curves defined over the rationals (or some other number field $K$). It is known that there are infinitely many elliptic curves with small rank (say rank at most 2), and that there exist elliptic curves over $\mathbb{Q}$ with rank as large as 28 (due to Elkies).

What is the largest positive integer $r$ such that it is known that there are infinitely many elliptic curves over the rationals with rank at least $r$?