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Claim. Theorem 2 of the paper is the statement of Problem 903: if $v=p^2+p+1$, then no 2-design (a family of blocks on $v$ points, every block having at least two points, in which every pair of points lies in exactly one block) has $b$ blocks with $p^2+p+1<b<p^2+2p+1$. So a block design on $n=p^2+p+1$ points with $t>n$ blocks has $t\ge n+p$, which is what the problem asks for $p$ a prime power; the theorem's hypothesis is only that $v$ has this form. The bound is sharp: replacing one line $\{x_1,\dots,x_{p+1}\}$ of a projective plane of order $p$ by the line $\{x_2,\dots,x_{p+1}\}$ and the $p$ pairs $\{x_1,x_i\}$ gives a 2-design with exactly $p^2+2p+1=n+p$ blocks. The lower bound $t\ge n$ for every such design with more than one block is the de Bruijn–Erdős theorem, on the 1948 source card, and a projective plane of order $p$ attains it.

The paper proves Theorems 2 and 3 twice, by a linear-algebraic argument on the incidence matrix and by a combinatorial one, and remarks that both follow from Totten's classification of the 2-designs with $(b-v)^2\le v$ by a longer route. Theorem 3 says that a design with $v=p^2+p+1$ points and exactly $p^2+2p+1$ blocks comes from a projective plane of order $p$ by replacing one line with a near pencil or a projective plane on its points, and Theorem 4 says that a design on these points that is not a projective plane, a near pencil or obtained from a projective plane by replacing one line has more than $p^2+(2+c)p$ blocks with $c=0.147899$; the site's commentary reports the latter as $t\ge n+cp$ with $c\approx1.148$. The source card records the paper's results, including Theorem 1 on the block counts a 2-design on $v$ points can have.

Depends on. No page of this wiki.

Acceptance. Published in the Journal of Combinatorial Theory, Series A 38 (1985), no. 2, 131–142 (refereed); the paper prints its receipt date, 1982-12-09, and the issue is dated March 1985 without a day, so the page is dated to the first day of that month. The site's curator, Thomas Bloom, lists Problem 903 as proved and credits it to Erdős, Fowler, Sós and Wilson [EFSW85] (reviewed). The reference was reported in the problem's forum thread on 14 October 2025 by a commenter who says it was located with GPT-5, and the site's page was updated to credit it. No Lean formalization is known; the formal-conjectures statement file, linked from the problem page, states the result as research solved with a sorry body and no formal_proof pointer, and a statement file is not a formalization.