Small Counterexamples to the All-n Form of the Exact Codegree Conjecture for Tight Hamiltonian Cycles

Rödl, Ruciński and Szemerédi stated the following conjecture, which they attribute to Katona and Kierstead: every k-uniform hypergraph on n ≥ k+1 ≥ 4 vertices in which every (k−1)-set lies in at least ⌊(n−k+3)/2⌋ edges has a tight Hamiltonian cycle. They proved it for k = 3 and all sufficiently large n. We observe that the statement, as written for all n ≥ k+1, fails for small n. The smallest counterexample is a 3-graph on 7 vertices: an apex joined to all pairs of a 6-set, together with the ten faces of the hemi-icosahedron on that set. It has minimum codegree 3 = ⌊7/2⌋ but no tight Hamiltonian cycle, and the proof is two lines. A variant of the construction gives counterexamples for k = 4 and k = 5 on k+4 vertices. A computer search finds two more, on 9 vertices for k = 3 and on 10 vertices for k = 5; the second meets the bound (n−k+3)/2 even without the floor. These are exceptions for small n only, and the asymptotic statement is not affected. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Scope: the note refutes the conjecture only in its all-n form. It does not concern the large-n statement, which was proved for k = 3 by Rödl, Ruciński and Szemerédi and has been announced for all k. Corpus identifiers: OWR-1782-009, OWR-1386-013. Paper page: https://eulersolve.org/papers/owr-1782-009/