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Claim. The corrected Statement of Problem 1069 holds: there is an absolute constant $c$ such that any $n$ points in the plane determine fewer than $c\,n^2/k^3$ lines containing at least $k$ of them, for every $2\leq k\leq n^{1/2}$. This is Theorem 2 of Szemerédi and Trotter, a short consequence of their Theorem 1, the incidence bound $c_1n^{2/3}t^{2/3}$ for $n$ points and $t$ lines when $n^{1/2}\leq t\leq\binom n2$: a family of $t$ lines each with at least $k$ points carries at least $kt$ incidences. Since $k\ge2$, each such line is determined by two of the points, so $t\le\binom n2$; if $t\ge n^{1/2}$, comparing the two bounds gives $t\le c_1^3n^2/k^3$, and if $t<n^{1/2}$, then $t<n^{1/2}\le n^2/k^3$ because $k\le n^{1/2}$. Either way $t\le\max(c_1^3,1)\,n^2/k^3$. The site's wording, whose range $k\leq n^{1/2}$ has no lower end, fails at $k=1$, as the problem page's Notes record; the theorem proves the corrected Statement in full. Erdős's 1987 problem paper (card, Section 2, p. 169) describes the statement as a conjecture of Croft, Purdy and Erdős and records this proof.

Acceptance. The paper is refereed: E. Szemerédi and W. T. Trotter, Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3-4, 381–392; the page is dated to the issue month, September 1983, which the publisher's record gives. Thomas Bloom, the site's curator, marks the problem solved and credits this paper on the problem's page at erdosproblems.com (page last edited 2025-10-02); that credit is the reviewed evidence. No Lean proof is recorded.

What remains. The best constant is unknown. At $k=n^{1/2}$ the lattice points give $(2+o(1))n^{1/2}$ lines each containing $n^{1/2}$ points, which Erdős thought might be extremal, and Sah's construction gives $(3+o(1))n^{1/2}$ such lines. The site cites it as Sah, Chih-Han, The rich line problem of P. Erdős (1987), 123–125, without a venue; Erdős's paper says the construction appears for the first time in the proceedings of the Siófok meeting, the volume holding Erdős's own paper.