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Dirac’s conjecture on the number of lines through a point in the plane

Prove Dirac’s conjecture asserting that, for any finite point set S ⊂ R^2 of affine dimension 2, the maximum number of distinct lines in S that pass through a single point satisfies maxlines(S) ≥ ⌊|S|/2⌋ + 1 for sufficiently large |S|.

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Background

To relate matroid minors to geometric incidence, the paper studies f(d), the fraction of points guaranteed to lie on many distinct lines through some point in a d-dimensional configuration. Strengthening lower bounds on this quantity directly improves column bounds for κ-bounded and Δ-modular matrices.

As context, the authors cite the classical Dirac conjecture in the planar case (d=2). Current best unconditional bounds (e.g., due to Han, building on Langer) guarantee only maxlines(S) ≥ ⌊|S|/3⌋ + 1, and the conjectured ⌊|S|/2⌋ + 1 remains open.

References

In particular, Dirac's conjecture states that for any real point set $S \subseteq 2$ of affine dimension $2$, that $\maxlines(S) \geq \lfloor |S|/2 \rfloor + 1$ for $|S|$ large enough.

— Excluding a Line Minor via Design Matrices and Column Number Bounds for the Circuit Imbalance Measure (2510.20301 - Dadush et al., 23 Oct 2025) in Our contributions — discussion of bounds on maxlines(S) (preceding Theorem thm:low-dim)