Smallest parameter admitting a discrete counterexample

Determine the smallest integer k for which the discrete orthogonal partition problem admits a counterexample, and in particular establish whether a counterexample exists for some integer k with 2≤k≤7.

Background

The discrete problem asks whether every finite planar point set with an even number of points admits two perpendicular lines producing cyclic class sizes k, k, m−k, and m−k, where the total number of points is 2m.

The paper proves that k=1 is always solvable and constructs a centrally symmetric 96-point counterexample for k=8. The smallest value of k at which failure can occur is therefore unknown, with the range 2≤k≤7 singled out explicitly.

References

The finite counterexample also leaves a basic quantitative problem open. Maldonado--Rold an-Pensado prove that $k=1$ is always solvable, while Theorem~\ref{thm:discrete-counterexample} gives a failure at $k=8$.

\begin{problem}\label{q:smaller-discrete} What is the smallest integer $k$ for which the discrete orthogonal partition problem admits a counterexample? In particular, does a counterexample exist for some $2\le k\le7$? \end{problem}

— Counterexamples and symmetry for uneven orthogonal mass partitions in the plane  (2609.16757 - Martínez-Sandoval, 15 Sep 2026) in Problem \ref{q:smaller-discrete}, Section ‘Discussion and further directions’, subsection ‘Remaining questions’