Everywhere unbalanced configurations
David Conlon, Jeck Lim
TL;DR
The paper studies a variant of Kupitz's problem on line-induced partitions, showing that one can force large imbalance using pseudolines by constructing generalised configurations with $n$ points where every pseudoline through a pair of points yields an imbalance of at least $k$, for any fixed $k$. It introduces a robust combinatorial toolkit—centred sequences, flips, blocks, shifting, and reflection—and a recursive step that amplifies a balance parameter $r$ via indicial sequences $(\alpha_i)$ and $(\beta_i)$. This yields a constructive pipeline that produces generalised configurations realizable as allowable sequences with $n \le 2^{2^{c k}}$ points, and hence tight up to a constant in the Pinchasi bound. The work also discusses stretchability and higher-dimensional analogues, highlighting open questions about realizability by straight lines and potential extensions.
Abstract
An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number $k$ such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most $k$. We give a negative answer to a natural variant of this problem, showing that for every natural number $k$ there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least $k$. Moreover, we may find such a configuration with at most $2^{2^{ck}}$ points, which, by a result of Pinchasi, is best possible up to the value of the constant $c$.