The Minimal Distance Problem

How well separated can a family of point-line pairs in the unit square be? For every $\varepsilon > 0$ there are arbitrarily large families $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$ with $x_i \in \ell_i$ and $\mathrm{dist}(x_i,\ell_j) \ge n^{-2/3-\varepsilon}$ for all $i \ne j$. Combined with earlier work of Cohen, Pohoata and Zakharov this settles the problem at the sharp exponent $2/3$. The same construction disproves a conjecture of Hunter, Pohoata, Verstraete and Zhang about induced point-line matchings over finite fields.