Erdős Problem #588

OPENdatabase updated 2025-08-31

No proof or disproof is known, and it isn't known whether a finite computation could settle it.

What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).

`numLinesWithPointsMax k n` is $f_k(n)$: the maximum number of lines containing exactly $k$ points among all sets $S$ of $n$ points in $\mathbb{R}^2$ with no $k + 1$ points on a line. Since $S$ has no $k + 1$ points on a line, the lines with at least $k$ points of $S$ are exactly the lines with exactly $k$ points of $S$. The definition follows Erdős Problem 101 (the case $k = 4$).

Let $f_k(n)$ be minimal such that if $n$ points in $\mathbb{R}^2$ have no $k+1$ points on a line then there must be at most $f_k(n)$ many lines containing at least $k$ points. Is it true that \[f_k(n) = o(n^2)\] for $k \geq 4$? The case $k = 4$ is Erdős Problem 101. The restriction $k \geq 4$ is needed: see `Erdos588.erdos_588.variants.three_not_isLittleO`.

Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 7 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/588.

Prize$100

Topicsgeometry

Statement in LeanFormal Conjectures

Solution in Leanno

OEISA006065, A008997

Original pageerdosproblems.com/588

Attempts

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WORKING REPORT BY THIS SITE · JUL 31, 2026

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Live page status is `OPEN - $100`, but “Currently working on this problem” lists `MrLaine` (accessed 2026-07-31); per the collision rule, no attempt was made.

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