Erdős Problem #903

The problem

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 $t\geq n+p$?

Statement quoted from erdosproblems.com/903, maintained by Thomas Bloom. Original source: [Er82e]. Current status there: proved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎