Erdős Problem #1069
SOLVEDdatabase updated 2025-10-05
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The \(k\)-rich lines of a finite set `s` of points in the plane: the lines of `ℝ²` which contain at least \(k\) points of `s`.
Given any \(n\) points in \(\mathbb{R}^2\), the number of \(k\)-rich lines (lines which contain \(\geq k\) of the points) is, provided \(k\leq n^{1/2}\), \[\ll \frac{n^2}{k^3}.\] Erdős [Er87b] describes this as a conjecture of himself, Croft, and Purdy. It was proved by Szemerédi and Trotter [SzTr83]. We take \(k \geq 2\), since for \(k \leq 1\) there are infinitely many \(k\)-rich lines.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/1069.Prizenone
Topicsgeometry
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1069
Attempts
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