  We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb
R}^3$, there are at most $O(N^{{3 \over 2}})$ points at which 3 lines intersect
non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$
lines in ${\Bbb R}^3$ so that no $N$ lines lie in the same plane and so that
each line intersects a set $P$ of points in at least $N$ points then the
cardinality of the set of points is $\Omega(N^3)$. Both our proofs are
adaptations of Dvir's argument for the finite field Kakeya problem.
