The board / graph theory / #600
Erdős Problem #600
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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Let \(e(n,r)\) be minimal such that every graph on \(n\) vertices with at least \(e(n,r)\) edges, each edge contained in at least one triangle, must have an edge contained in at least \(r\) triangles.
Let \(r \geq 2\). Is it true that \(e(n,r+1) - e(n,r) \to \infty\) as \(n \to \infty\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/600.Prizenone
Topicsgraph theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/600
Attempts
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exact e(n,r) is verified through n=10 and closed forms are proved for r=n-2 and r=n-3, but both fixed-r asymptotic questions remain open.
PARTIAL