Combinatorial Theory

An \(r\)-uniform tight cycle of length \(\ell>r\) is a hypergraph with vertices \(v_1,\dots,v_\ell\) and edges \(\{v_i,v_{i+1},\dots,v_{i+r-1}\}\) (for all \(i\)), with the indices taken modulo \(\ell\). It was shown by Sudakov and Tomon that for each fixed \(r\geq 3\), an \(r\)-uniform hypergraph on \(n\) vertices which does not contain a tight cycle of any length has at most \(n^{r-1+o(1)}\) hyperedges, but the best known construction (with the largest number of edges) only gives \(\Omega(n^{r-1})\) edges. In this note we prove that, for each fixed \(r\geq 3\), there are \(r\)-uniform hypergraphs with \(\Omega(n^{r-1}\log n/\log\log n)\) edges which contain no tight cycles, showing that the \(o(1)\) term in the exponent of the upper bound is necessary.

Mathematics Subject Classifications: 05C65, 05C38