Araujo-Piga-Schacht Question on Tight Hamilton Cycles

Araujo, Piga and Schacht asked whether density and codegree both above $1/4$ force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is $p_0 = \max_{0 \le x \le 1}\min\{x^3, 1-x\} \approx 0.3177$ , and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every $p > 1/3$ the asymptotically sharp minimum-codegree threshold is determined.

Result: Disproved(see note)
Status: Resolved
AI contribution: AI-assisted
Method: Construction
Field: Hypergraph theory
Posed by: Araujo, Piga, Schacht
Year posed: —
Years open: —
Solved: 2026-07-23
Model: ChatGPT
Vendor: OpenAI
Collaborators: Xichao Shu
Verification: Unreviewed
Publication: Preprint
Significance: 15 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

the question is answered negatively and the correct threshold is determined

What the AI did

The author acknowledges using ChatGPT in the early stage of the project and during manuscript preparation, and states specifically that an initial idea leading to the first construction in the paper arose during an interaction with it.

Verification

Single-author arXiv preprint; not yet peer-reviewed.

Source

arXiv:2607.21568 - Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs

Araujo-Piga-Schacht Question on Tight Hamilton Cycles

What was actually shown

What the AI did

Verification

Source

Discussion