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Claim. Theorem 4 of Szemerédi and Trotter [SzTr83] proves that the number $\mathscr E(n)$ of distinct nondecreasing sequences $y_1\le\cdots\le y_t$ for which some set $P$ of $n$ points in the plane and some lines $\ell_1,\dots,\ell_t$ determined by $P$ have $\lvert\ell_j\cap P\rvert=y_j$ for every $j$ satisfies $\mathscr E(n)<2^{c_4\sqrt n}$ for all $n\ge1$, with an absolute constant $c_4$; the paper records that this settles a conjecture of Erdős. The quantity $F(n)$ of Problem 607 counts the distinct sets $A$ of line sizes rather than the sequences. Each such set is the set of values of one of the sequences counted by $\mathscr E(n)$, so $F(n)\le\mathscr E(n)\le\exp(O(\sqrt n))$, which answers the question affirmatively. That last step is a one-line remark recorded on the source card, not a statement of the paper; the site credits the paper with the proof directly. The site's commentary also reports Erdős's view that the bound is best possible, which is not part of this claim.

Acceptance. The site's curator, T. F. Bloom, labels the problem PROVED and credits Szemerédi and Trotter [SzTr83] (problem page accessed), which is the reviewed evidence. The paper is refereed: E. Szemerédi and W. T. Trotter, Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3–4, 381–392, DOI 10.1007/BF02579194, received 1982-08-19 and revised 1983-03-14; the page is dated by the issue month, September 1983, on the first of the month. The source card digests the paper and records the statement of Theorem 4. No independent proof review and no formalization are recorded.