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Claim. The paper's abstract states its theorem thus: "a pairwise balanced design on $n$ points in which each block is of size at least $n^{1/2}-c$ can be embedded in a projective plane of order $n+i$ for some $i\le c+2$ if $n$ is sufficiently large", and adds that "if the projective plane conjecture is true, the conjecture of Erdős and Larson will not be true." The abstract writes $n$ both for the number of points and inside the plane's order, where it cannot mean the number of points: a plane of order $k$ has $k^2+k+1$ points and lines of $k+1$ points, so a design on $n$ points whose blocks have at least $n^{1/2}-c$ points lies in no plane of order below $n^{1/2}-c-1$. Read with the problem's $n$, the number of points, the theorem says: for every $c>0$ there is $n_0(c)$ such that every pairwise balanced design on $n\ge n_0(c)$ points with every block of size at least $n^{1/2}-c$ embeds in a projective plane whose order $k$ satisfies $k\le n^{1/2}+c+2$; together with the lower bound just noted, $k$ lies within $c+2$ of $n^{1/2}$. The zbMATH review (Zbl 0617.05013) records the result as the embedding of certain pairwise balanced designs in a projective plane, from which the Erdős–Larson conjecture is false if the projective plane conjecture holds; Erdős restates it the same way, with the abstract's wording, as Problem 8 (p. 3) of Some unsolved problems [Er97f]. The theorem's numbering and proof inside the paper are not recorded here.

The deduction: if a constant $C$ and designs on every large $n$ with $|A_i|>n^{1/2}-C$ for all $i$ existed, as Problem 665 asks, the theorem would give a projective plane with order in the window $[n^{1/2}-C-1,\,n^{1/2}+C+2]$ for every large $n$, so every window of length $2C+3$ far enough out would contain the order of a projective plane. Prime powers have arbitrarily long gaps (the gaps between consecutive primes are unbounded, and the proper prime powers are too sparse to fill them), so under the hypothesis infinitely many such windows contain no prime power and no plane, and the answer to the problem is no. The Erdős–Larson conjecture that the paper names is this question, posed in Erdős and Larson 1982, whose Theorem 1 gives designs with $|A_i|=n^{1/2}+O(n^{1/2-c})$ for an absolute $c>0$.

Hypothesis. The unproven conjecture that every finite projective plane has prime power order, which is Problem 723: planes of every prime power order exist, and no plane of another order is known, but the conjecture is open even for order $12$. Without it the theorem says only that a design as the problem asks forces a projective plane of an order within $C+2$ of $n^{1/2}$, and the problem stays open.

Acceptance. Refereed: S. S. Shrikhande and N. M. Singhi, On a problem of Erdős and Larson, Combinatorica 5 (1985), no. 4, 351–358, received 21 June 1983 and issued December 1985, the date this page is named by; the day is a placeholder for the issue month. The site's credit on a problem it labels OPEN is not reviewed evidence.