  We prove that any ergodic $SL_2(R)$-invariant probability measure on a
stratum of translation surfaces satisfies strong regularity: the measure of the
set of surfaces with two non-parallel saddle connections of length at most
$\epsilon_1, \epsilon_2$ is $O(\epsilon_1^2 \epsilon_2^2)$. We prove a more
general theorem which works for any number of short saddle connections. The
proof uses the multi-scale compactification of strata recently introduced by
Bainbridge-Chen-Gendron-Grushevsky-M\"oller and the algebraicity result of
Filip.
