Problem 210

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?

Second statement

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?

Source: erdosproblems.com/210

Geometry

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SolvedProved

An accepted solution exists. The statement is true.

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.

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.

Problem 210

Second statement

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