We show that the class of monotone $2^{O(\sqrt{\log n})}$-term DNF
formulae can be PAC learned in polynomial time under the uniform
distribution. This is an exponential improvement over previous
algorithms in this model, which could learn monotone
$o(\log^2 n)$-term DNF, and is the first efficient algorithm
for monotone $(\log n)^{\omega(1)}$-term DNF in any nontrivial
model of learning from random examples. Our result extends to any
constant-bounded product distribution.