Araujo-Piga-Schacht Question on Tight Hamilton Cycles
Prior state unknown→disproved
the question is answered negatively and the correct threshold is determined
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatorics / Hypergraph theory
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.
15Significance / 100
1Frontier events
0Verification tasks
0Recorded attempts
Temporal state
Current frontier
No reconciled state yet.
Append-only history
Frontier timeline
Research memory
Claims and attempts
Scoped claims
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.
the question is answered negatively and the correct threshold is determined
Recorded attempts
Evidence graph
Connected research record
No public relationships recorded yet.