Problem 903

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claims/: The 1 claim page of Problem 903, one per claimant's result; the problem's standing derives from them.

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Statement. Let $n=p^2+p+1$ for some prime power $p$, and let $A_1,\ldots,A_t\subseteq \{1,\ldots,n\}$ be a block design (so that every pair $x,y\in \{1,\ldots,n\}$ is contained in exactly one $A_i$).

Is it true that if $t>n$ then $t\geq n+p$?

Formulation. The site's parenthetical gives only the pair condition. A block design is read as Erdős, Fowler, Sós and Wilson define a 2-design [EFSW85, p. 131], with every block of at least two points. If empty or one-point blocks were allowed, the wording would fail trivially, since a projective plane of order $p$ plus one singleton block has $n+1$ blocks. The page's standing concerns the reading with blocks of at least two points.

Status. Proved. The site labels the problem PROVED (page last edited 24 October 2025).

Source. erdosproblems.com/903, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #903, https://www.erdosproblems.com/903.

References.

• [EFSW85] Erdős, P. and Fowler, Joel C. and Sós, Vera T. and Wilson, Richard M., On $2$-designs. J. Combin. Theory Ser. A 38 (1985), no. 2, 131--142.
• [dBEr48] de Bruijn, N. G. and Erdős, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. 51 (1948), 1277--1279 = Indagationes Math. 10, 421--423.

Formalization. Statement in formal-conjectures, added on 2026-10-07. Its erdos_903 states the question with the answer yes, tagged research solved with a sorry body and no formal_proof pointer; a statement file is not a formalization of a result.

Current assessment

The question, in the site's formulation of 2026-09-04, is a conjecture of Erdős and Sós: a block design on $n=p^2+p+1$ points, $p$ a prime power, with more than $n$ blocks has at least $n+p$ of them. It is proved. The de Bruijn–Erdős theorem [dBEr48] gives $t\ge n$ for every such design with more than one block, with equality for a projective plane of order $p$, and Theorem 2 of Erdős, Fowler, Sós and Wilson [EFSW85] excludes every block count strictly between $n$ and $n+p$, for every $v=p^2+p+1$ and not only for prime powers. The accepted claim page Erdős, Fowler, Sós and Wilson 1985 states the theorem, the construction attaining $n+p$, the two proofs and the acceptance evidence, a refereed journal paper credited by the site's curator. The same paper classifies the designs with exactly $n+p$ blocks (Theorem 3) and shows (Theorem 4) that every design on these points that is not a projective plane, a near pencil, or obtained from a projective plane by breaking up one line has more than $p^2+(2+c)p$ blocks, with $c=0.147899$. The site's commentary asks, more generally, which block counts a design on $n$ points can have, which the paper's Theorem 1 answers for $v>v_0$ and all but an interval just above $n$.

Search scope, 2026-10-07: the site's problem page and discussion thread and the formal-conjectures statement file. No claim other than the paper's was found.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• bruijn_1948_combinatorial_problem
• bruijn_1948_combinatorial_problem / theorem_1
• erdos_1985_2_designs
• erdos_1985_2_designs / theorem_1
• erdos_1985_2_designs / theorem_2
• erdos_1985_2_designs / theorem_3
• erdos_1985_2_designs / theorem_4