Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 210

claims/: The 4 claim pages of Problem 210, one per claimant's result; the problem's standing derives from them.

Statement. Let $f(n)$ be minimal such that the following holds. For any $n$ points in $\mathbb{R}^2$ , not all on a line, there must be at least $f(n)$ many lines which contain exactly 2 points (called 'ordinary lines'). Does $f(n)\to \infty$? How fast?

Statement (corrected). Let $f(n)$ be maximal such that the following holds. For any $n$ points in $\mathbb{R}^2$ , not all on a line, there must be at least $f(n)$ many lines which contain exactly 2 points (called 'ordinary lines'). Does $f(n)\to \infty$ ? How fast?

Notes. The site writes "minimal". Read as the site words it, every $f$ up to the least number of ordinary lines has the stated property, so the minimal such $f$ is $0$ and the first question fails trivially. The site's commentary means the least number of ordinary lines spanned by $n$ points in the plane, not all on a line: it calls $f(n)\ge1$ the Sylvester-Gallai theorem and credits Motzkin, Kelly and Moser, Csima and Sawyer, and Green and Tao with lower bounds for that quantity. That quantity is the largest $f$ with the property, so the only change is "minimal" to "maximal". The claim pages are scoped against this Statement.

Status. Proved.

Source. erdosproblems.com/210, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #210, https://www.erdosproblems.com/210.

References.

[CsSa93] Csima, J. and Sawyer, E. T., There exist $6n/13$ ordinary points. Discrete Comput. Geom. (1993), 187-202.
[GrTa13] Green, Ben and Tao, Terence, On sets defining few ordinary lines. Discrete Comput. Geom. (2013), 409-468.
[KeMo58] Kelly, L. M. and Moser, W. O. J., On the number of ordinary lines determined by $n$ points. Canadian J. Math. (1958), 210-219.
[Mo51] Motzkin, Th., The lines and planes connecting the points of a finite set. Trans. Amer. Math. Soc. (1951), 451-464.

Formalization. None recorded.

Current assessment

The question has two parts: whether the least number $f(n)$ of ordinary lines (lines through exactly two of the points) determined by $n$ points in the plane, not all on one line, tends to infinity, and how fast it grows. The Sylvester-Gallai theorem, conjectured by Sylvester in 1893, rediscovered by Erdős in 1933 and proved by Gallai, gives $f(n)\ge 1$ ; the growth question is due to Erdős and de Bruijn.

Both parts are answered, by four accepted claims. Motzkin (1951) proves $f(n)\to\infty$ , with a bound of order $\sqrt n$ . Kelly and Moser (1958) prove $f(n)\ge 3n/7$ for every $n$ , sharp at $n=7$ . Csima and Sawyer (1993) prove $f(n)\ge 6n/13$ for $n\ge 8$ . Green and Tao (2013) determine $f(n)$ exactly for all large $n$ (Theorem 2.2 of their paper): $f(n)=n/2$ for even $n$ , attained by half the points equally spaced on a circle and half at infinity, and $f(n)=3\lfloor n/4\rfloor$ for odd $n$ , attained by the Böröczky examples. The bound $n/2$ is the Dirac-Motzkin conjecture. The site says that Motzkin conjectured $n/2$ for $n\ge 13$ ; Green and Tao note that neither Dirac nor Motzkin seems to have conjectured it formally in print (Dirac twice calls it likely, and Motzkin does not seem to mention it), and the Crowe-McKee configuration of $13$ points with $6$ ordinary lines shows that the bound $n/2$ fails at $n=13$ . An earlier claimed proof of the $n/2$ bound for large $n$ , by Hansen, is recorded on the site as flawed. The frontmatter standing derives from the Green-Tao full claim; the three earlier claims are partial. All four results are refereed, and the site's curator credits each of them.

The corpus holds no proof review of these results and does not hold the Motzkin and Csima-Sawyer papers. The exact value of $f(n)$ for small $n$ lies outside the question.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (22)

The number of ordinary lines grows without bound At least 3n/7 ordinary lines for every n At least 6n/13 ordinary lines for n at least 8 At least n/2 ordinary lines for large n problems/discrete_geometry/E0210/claims Discrete and Convex Geometry Discrete and Convex Geometry discrete_geometry/erdos_1984_research_problems discrete_geometry/green_2013_sets_defining_few_ordinary_lines Proposition 2.1 (p. 10): the Böröczky examples and their ordinary-line counts Theorem 1.2 (p. 2): n points not all on a line span at least n/2 ordinary lines for large n Theorem 1.5 (p. 6): full structure theorem for sets with at most Kn ordinary lines Theorem 2.2 (p. 13): the exact least number of ordinary lines for large n, with its extremisers Theorem 2.4 (p. 14): strong Dirac-Motzkin, sets with at most n - C ordinary lines are Böröczky-type discrete_geometry/kelly_1958_number_ordinary_lines_determined_points Inequality 4.5 (p. 215): t_2 >= 3 + t_4 + 2t_5 + 3t_6 + ...Theorem 3.6 (p. 213): n points not all on a line determine at least 3n/7 ordinary lines distance_problems/erdos_1983_combinatorial_problems_geometry Ordinary lines (pp. 38--39): de Bruijn and Erdős's conjecture, Motzkin's theorem, Kelly and Moser's 3n/7 and Motzkin's n/2 conjecture set_systems/bruijn_1948_combinatorial_problem Remark (p. 422): whether the least number f(n) of two-point lines tends to infinity Gallai's theorem (p. 421): n points of the plane, not all on a line, span a line through exactly two of them

Graph