For every d>0 and every alpha>0, do there exist rho>0 and n_0 such that every (rho,d)-dense 3-uniform hypergraph H on n≥ n_0 vertices with minimum codegree delta_2(H)≥ (frac13+alpha)n contains a tight Hamiltonian cycle? Here (rho,d)-dense means that every vertex subset U⊆ V(H) with |U|≥ rho n satisfies e(H[U])≥ d binom|U|3, and a tight Hamiltonian cycle is a cyclic ordering of V(H) such that every three consecutive vertices form an edge of H.

Aigner–Horev–Levy problem and Araújo–Piga–Schacht question on tight Hamiltonian cycles

Aigner–Horev–Levy’s problem asks when a sufficiently large uniformly dense $3$ -graph must contain a tight Hamiltonian cycle; Araújo–Piga–Schacht asked whether density $1/4$ could make a codegree threshold near $n/4$ sufficient. Chen, Han, and Liu claim definitive answers in a stronger form.

Known results

Aigner–Horev and Levy proved an analogous existence theorem under the strongest density notion; the retrieved source gives no year or full threshold.
A prior result established the asymptotically sharp unrestricted minimum vertex-degree threshold at $\frac59\binom{n}{2}$ for tight Hamiltonian cycles.
Related work records the earlier codegree-based tight-cycle results and motivates stronger uniform-density hypotheses.

August 2026 claimed resolution

The preprint claims that minimum codegree at least $(\frac13+\alpha)n$ forces a tight Hamiltonian cycle in sufficiently large uniformly dense $3$ -graphs, with counterexamples below $\frac13 n$ . It also gives uniformly dense examples near $\frac59\binom n2$ without such a cycle, answering the Araújo–Piga–Schacht question negatively. Two related July preprints independently claim matching sharp-threshold results, but no verification or referee report was found.

Current status (as of August 2026): The claimed thresholds and counterexamples appear to resolve both questions, but the resolution remains unverified because the central evidence is an unrefereed preprint.

Aigner–Horev–Levy problem and Araújo–Piga–Schacht question on tight Hamiltonian cycles

Progress summary

Primary source

Additional references

Solutions 0