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 $r$?



Largest rank assumed by infinitely many elliptic curves
As suggested by @JoeSilverman, I am turning my comment into an answer.


According to the table maintained by Dujella at
<http://web.math.pmf.unizg.hr/~duje/tors/generic.html>,
the current record is **19**,
due to Noam Elkies in 2006. This is apparently obtained by a family of 
elliptic curves over an elliptic curve over $\mathbb Q$ with positive rank
such that the generic fiber has Mordell-Weil group of rank 19. By a theorem
of Neron (or a stronger version due to Silverman [Thm. C in "Heights and
the specialization map for families of abelian varieties", J. Reine Angew.
Math. 342 (1983), 197-211]), for all but finitely many of the infinitely
many rational points of the base elliptic curve, the specialization map
from the Mordell-Weil group of the generic fiber to that of the fiber
above that point is injective, so that the rank of the latter is also
at least 19.


