Common Neighbour Conjectures for Saxl Graphs
The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size and its later generalisation to arbitrary base size. For every integer , it constructs infinitely many primitive permutation groups of base size whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size , it gives infinite counterexample families of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types.
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Occurred: Sep 1, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: Common Neighbour Conjectures for Saxl Graphs · Saxl Common Neighbour
Confidence: Not scored
Registry verification: unreviewed · preprint · resolved
Attribution
VibeMathed
registry · event recorded by
ChatGPT Pro
model · ai model contributor · OpenAI
Claude
model · ai model contributor · Anthropic
Aluna Rizzoli
human · human collaborator
Adam R. Thomas
human · human collaborator
Lineage and corrections
This event attributed to ChatGPT Pro
This event attributed to Aluna Rizzoli
VibeMathed record: Common Neighbour Conjectures for Saxl Graphs evidence for this event
This event attributed to Claude
This event attributed to Adam R. Thomas
Common Neighbour Conjectures for Saxl Graphs evidence for this event
The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size and its later generalisation to arbitrary base size. For every integer , it constructs infinitely many primitive permutation groups of base size whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size , it gives infinite counterexample families of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types. parent of this event
Common Neighbour Conjectures for Saxl Graphs parent of this event
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