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Beck proves that there is an absolute constant $c>0$ such that $n$ points in the real plane with at most $n-k$ of them on any line determine at least $ckn$ distinct lines, each through at least two of the points. This is the statement of Problem 211, which Erdős had conjectured. In particular $2n$ points with at most $n$ on a line determine $\gg n^2$ lines. Erdős records that Beck proved it in his 1984 problem note, whose source card [[../library/discrete_geometry/erdos_1984_research_problems/_index|states the result as Beck's theorem]], and adds that Beck's value of $c$ seems too small; the sharper question of whether $c=1/6$ works is discussed on the problem page. The same bound also follows from the incidence theorems of Szemerédi and Trotter, published in the same issue, as Erdős notes in the 1984 note; their paper does not state it.

The result is refereed: József Beck, On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry, Combinatorica 3 (1983), no. 3-4, 281-297. The site's curator, T. F. Bloom, marks the problem proved and credits this paper, and Erdős himself, in the 1984 note, records that Beck proved his conjecture. The paper is not held.