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.

翻译：我们给出了单位立方体中点集的最小离差及其反函数在高维情形下的一个新下界。通过仅考虑非常小的一类测试盒子，我们将最小离差的界化问题简化为极值集合论中的一个问题。具体而言，我们将$r$-无覆盖族大小的下界转化为点集最小离差反函数的下界。我们得到的下界与最近获得的最小离差上界在相差对数项的意义上是匹配的。