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Claim. Let $g(n)$ be the largest number such that every set of $n$ points in $\mathbb{R}^2$ with no four on a line contains $g(n)$ points with no three on a line, the function of Problem 589 (the paper's $\alpha(n)$). Theorem 2.1 of J. Balogh and J. Solymosi, On the number of points in general position in the plane, states that, as $n\to\infty$,

$$g(n)\le n^{5/6+o(1)}:$$

there are $n$-point planar sets with no four points on a line in which every subset of $n^{5/6+o(1)}$ points contains three collinear points. The construction is a subset of the grid $[m]^3$ projected generically to the plane, and the proof replaces the density Hales--Jewett route of Füredi by the hypergraph container method of Balogh, Morris and Samotij and of Saxton and Thomason, applied to the $3$-uniform hypergraph of collinear triples with a supersaturation lemma for large subsets of the grid. The authors write that it is far from clear whether $5/6$ is the right exponent. The [[../library/discrete_geometry/balogh_2018_number_points_general_position_plane/_index|source card]] records the paper's results, including its $\varepsilon$-net theorems, which concern other problems.

Covers. The upper bound $g(n)\le n^{5/6+o(1)}$ only. Not covered: any lower bound, and the order of $g(n)$, which is not determined; the lower bound $c\sqrt{n\log n}$ remains Füredi's.

Depends on. Nothing in this wiki; the claim rests on the cited paper and the container theorem it applies.

Acceptance. Refereed: J. Balogh and J. Solymosi, On the number of points in general position in the plane, Discrete Analysis 2018:16, 20 pp., received 1 September 2016 and revised 17 April 2017 as the paper prints. The site's commentary records the bound, but the site labels the problem OPEN, so that remark is not acceptance of the problem and the page lists no reviewed evidence. The proof is not compiled in this corpus.

Dating. The page is dated by the first public posting, arXiv:1704.05089 of 17 April 2017. The journal's record dates the publication 13 September 2018; the paper's own front matter prints 12 October 2018.