  We give a new lower bound for the minimal dispersion of a point set in the
unit cube and its inverse function in the high dimension regime. This is done
by considering only a very small class of test boxes, which allows us to reduce
bounding the dispersion to a problem in extremal set theory. Specifically, we
translate a lower bound on the size of $r$-cover-free families to a lower bound
on the inverse of the minimal dispersion of a point set. The lower bound we
obtain matches the recently obtained upper bound on the minimal dispersion up
to logarithmic terms.
