Solymosi's general-position conjecture for rich lines in grids

Let $\mathbb{F}$ be a field, let $0<\alpha\leq 1$ , and let a line $\ell$ in $\mathbb{F}^2$ be $\alpha$ -rich in a Cartesian product $Y\times Y$ if

|\ell\cap (Y\times Y)|\geq \alpha|Y|.

A set of lines is in general position if no two lines are parallel and no three lines pass through a common point. Solymosi's conjecture. Among the lines that are $\alpha$ -rich in a $N\times N$ Cartesian product, at most $C=C(\alpha)>0$ can be in general position. The conjecture concerns the inverse problem for rich-line incidences: it asserts that, absent structure among the lines, a grid supports only boundedly many rich lines. The paper disproves it with explicit examples over $\mathbb{Q}$ , $\mathbb{C}$ , and $\mathbb{F}_p$ , so the conjecture is refuted.

The conjecture predicts a bound depending only on $\alpha$ for rich lines in general position in an $N\times N$ grid. A September 2017 paper claims to refute it by constructing unbounded families.

Known results

October 2013: over $\mathbb{R}$ , sufficiently many lines with at least $n^{1-\delta}$ grid points force two parallel lines or three concurrent lines; this is weaker than a constant bound.
The same 2013 work gives the equivalent formulation for arbitrary $A\subseteq\mathbb{R}$ with $|A|=n$ .

September 2017 counterexample

The paper Upper and lower bounds for rich lines in grids claims that, for fixed $0<\alpha<1$ , over $\mathbb{Q}$ the number of rich lines in general position is at least $C(1-\alpha)\log N/\log\log N$ . It also gives explicit constructions over $\mathbb{C}$ and $\mathbb{F}_p$ , thereby claiming a refutation in all stated settings.

Current status (as of September 2026): The conjecture has a claimed explicit refutation from 2017; no later source in the record overturns it, but this automated summary does not independently verify the proof.

Solymosi's general-position conjecture for rich lines in grids

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