The board / graph theory / #794
Erdős Problem #794
DISPROVED LEANdatabase updated 2026-02-05
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).
The `3`-uniform hypergraph on `Fin 9` given by Harris: the `27` triples containing exactly one vertex from each of the three parts `{0, 1, 2}`, `{3, 4, 5}`, `{6, 7, 8}`, together with the triple `{0, 1, 2}`. A triple is a transversal of the three parts exactly when the three values of `v ↦ v / 3` it produces are distinct.
Is it true that every \(3\)-uniform hypergraph on \(3n\) vertices with at least \(n^3+1\) edges must contain either a subgraph on \(4\) vertices with \(3\) edges or a subgraph on \(5\) vertices with \(7\) edges? Harris has provided the following simple counterexample to the problem as stated: the \(3\)-uniform graph on \(\{1,\ldots,9\}\) with \(28\) edges, formed by taking \(27\) edges by choosing one element each from \(\{1,2,3\},\{4,5,6\},\{7,8,9\}\), and then adding the edge \(\{1,2,3\}\).
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/794.Prizenone
Topicsgraph theory, hypergraphs, turan number
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/794
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.