Proved
Established result in cited source ↗
Counting flags in triangle-free digraphs · Theorem 1.2; §2 notation; §5 computation →
Observed Oct 12, 2026
Bound · version 1
A proved 0.3465 triangle threshold
Statement
Every finite oriented graph on n ≥ 1 vertices with minimum out-degree at least 0.3465n contains a directed triangle.
Assumptions
- The assumption is stronger than n/3. This published threshold is selected evidence, not a claim to be the best current bound.
Evidence
Source reports
Connections
In this problem · publication v9 · Selected context
Caccetta–Häggkvist cycle bound →
Bound · Proved
Scope: The directed triangle conclusion holds at degree 0.3465n.
Limit: The intended threshold n/3 is lower and remains unresolved.
These results can cover the same graphs; their coverage is not added or expressed as a percentage.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v11
Caccetta–Häggkvist cycle bound →
Bound · Proved
Scope: The directed triangle conclusion holds at degree 0.3465n.
Limit: The intended threshold n/3 is lower and remains unresolved.
These results can cover the same graphs; their coverage is not added or expressed as a percentage.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Counting flags in triangle-free digraphs →
- Theorem 1.2; §2 notation; §5 computation → · Original passage ↗
Recorded wording
An oriented graph with minimum out-degree at least 0.3465 times its order contains a directed triangle. The paper forbids opposite arcs and uses flag algebras with auxiliary arithmetic.
paraphrase · Follow the original source for authoritative statements and reuse terms.
Jan Hladký · Daniel Král’ · Sergey Norin · Published · Publication date 2017
Source-backed selection; publication and proof assessment are separate.
Revision & attribution
Source revision: arxiv-0908.2791v4. Observed Oct 12, 2026.
AI use is not reported in the selected source.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Established result in cited source ↗
Observed Oct 12, 2026 · Source passage →
Record details
- Claim ID
- claim-ch-triangle-03465
- Statement version
- 1
- SHA-256
- 5b445e43f64f0eee91032188a158e0ae74df6e27da3f3ca8df9a7c1af4c8300b