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：我们给出了高维情形下单位立方体中点集的最小分散及其逆函数的一个新下界。该结果通过仅考虑一类非常小的测试框而获得，这使我们能够将分散的界化归为极值集合论中的一个问题。具体而言，我们将关于$r$-覆盖自由族规模的下界转化为点集最小分散的逆函数的下界。所得下界与近期获得的最小分散上界在对数项范围内一致。