Dirac's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite non-collinear set $P\subset\mathbb R^2$ with $n=|P|\geq3$, let $\mathcal L(P)$ be the set of distinct affine lines containing at least two points of $P$, and let $d_P(p)=|\{\ell\in\mathcal L(P):p\in\ell\}|$.

Statement recorded in the supplied source.

Is there an absolute real constant $c\geq0$, independent of $n$ and $P$, such that \[\forall P\subset\mathbb R^2\text{ finite and non-collinear},\qquad\max_{p\in P}d_P(p)\geq \frac{|P|}{2}-c?\] [S2]

Short English statement

Must some point of every non-collinear planar configuration lie on at least half as many determined lines as there are points, up to one fixed additive constant?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3089 (OPG-59888). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Dirac, Gabriel, Dirac's Conjecture, Open Problem Garden, node 59888. Complete problem, discussion and bibliography (including mathematical image alt text). https://www.openproblemgarden.org/op/diracs_conjecture.