The board / combinatorics / #734
Erdős Problem #734
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Find, for all large \(n\), a non-trivial pairwise balanced block design \(A_1,\ldots,A_m\subseteq \{1,\ldots,n\}\) such that, for all \(t\), there are \(O(n^{1/2})\) many \(i\) such that \(\lvert A_i\rvert=t\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/734.Prizenone
Topicscombinatorics
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/734
Attempts
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exact f(n)=(3,3,4,4,6,6,6,6,6) for 3<=n<=11 with a from-scratch exhaustive verifier; proved an Omega(n^(3/4)) projective-truncation obstruction and isolated a quantifier gap in the current conditional
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