Borel n-point sets in the plane for fixed n ≥ 2

Determine, for each fixed integer n ≥ 2, whether there exists a Borel subset M ⊆ R^2 that intersects every line l in exactly n points.

Background

Mazurkiewicz (two-point) sets are classical choice-based constructions where a set meets every line in a prescribed number of points. Mauldin studied the Borel feasibility when the number of intersection points varies with the line, showing the function l ↦ α_l must be Borel.

While there are simple Fσ examples for countably many intersection points, the case of a fixed finite number n ≥ 2 of intersection points on every line remains unresolved in the Borel context.

References

Nevertheless, the case of $\alpha_l$ being a fix natural number $n$ for any $n\geq2$ is still open.

— Partitions of $\mathbb{R}^3$ into unit circles with no well-ordering of the reals  (2501.03131 - Fatalini, 6 Jan 2025) in Subsubsection “Mazurkiewicz sets” (Subsection \ref{subsection: lit review paradoxical sets})

Is there a Borel two-point set?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 5