The board / graph theory / #775
Erdős Problem #775
DISPROVED LEANdatabase updated 2026-04-19
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Is there a \(3\)-uniform hypergraph on \(n\) vertices which contains at least \(n-O(1)\) different sizes of cliques (maximal complete subgraphs)? The answer is no, as proved by Gao [Ga25]: more generally, for any \(k\geq 3\), every \(k\)-uniform hypergraph on \(n\) vertices contains at most \(n-f_k(n)\) different sizes of cliques, where \(f_k(n)\to \infty\) as \(n\to \infty\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/775.Prizenone
Topicsgraph theory, hypergraphs
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/775
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.