Counting flags in triangle-free digraphs
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The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
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.
Passage ID: caccetta-haggkvist:entry:triangle-bound:v1
Source item: Theorem 1.2; §2 notation; §5 computation
Source record: hkn-triangle-bound@arxiv-0908.2791v4
The triangle specialization asks for a directed triangle at minimum out-degree at least one third of the vertex count, in an oriented graph.
Passage ID: caccetta-haggkvist:entry:triangle-target:v1
Source item: §1, triangle specialization
Source record: hkn-triangle-bound@arxiv-0908.2791v4
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v11
Source assessment
Proved
Established result in cited source
The assumption is stronger than n/3. This published threshold is selected evidence, not a claim to be the best current bound.
Observed Oct 12, 2026
Source assessment
Open
Open question in cited source
Oriented graphs have no loops, parallel arcs or opposite arcs; this is a specialization of the full conjecture.
Observed Oct 12, 2026
Research activity
Arithmetic accompanying the triangle bound
The authors provide companion Maple arithmetic supporting the published argument.
An external resource link is not an OpenMath execution or formal verification of the full conjecture.
Question
Caccetta–Häggkvist cycle bound
Connection
The one-third triangle question is a specialization for oriented graphs.
Further reading
Read the mathematical argument; inspect its Claim for assessment and scope.
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Project notes
No OpenMath execution recorded
Companion arithmetic for the published bound; no OpenMath execution is attached.
Resource revision: 0908.2791v4
Recorded authorship
Jan Hladký · Daniel Král’ · Sergey Norin
Jan Hladký; Daniel Král’; Sergey Norin
AI use is not reported in the selected source.
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Counting flags in triangle-free digraphs
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