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 \(n\ge k+1\ge 4\) vertices in which every \((k-1)\)-set lies in at least \(\lfloor (n-k+3)/2\rfloor\) 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 \(n\ge k+1\), 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 \(3=\lfloor 7/2\rfloor\) 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.