Existence of k‑everywhere‑unbalanced point sets for k ≥ 3

Determine whether, for each integer k ≥ 3, there exists a finite point set S ⊂ ℝ^2 that is k‑everywhere‑unbalanced, meaning that for every line through two points of S, the absolute difference between the numbers of points on the two sides of the line is at least k.

Background

A set S of n points is called k‑everywhere‑unbalanced (k‑EU) if every line defined by two points of S has imbalance at least k, i.e., the difference in counts of points on either side is at least k. Kupitz asked about existence for k=2, and Alon proved 2‑EU sets exist for infinitely many even sizes (with s‑fold rotational symmetry) and gave explicit constructions. Pinchasi showed any k‑EU set, if it exists, requires Ω(2^{2^{k}}) points, and Conlon–Lim gave near‑matching upper bounds in the pseudoline setting. The authors of the present paper give SAT‑aided evidence and realizations, including a minimal odd‑sized 21‑point 2‑EU set, but the existence for k ≥ 3 remains open.