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

A family \(\mathcal{F}\) of subsets of \(\{1,\dots,n\}\) is called \(k\)-wise intersecting if any \(k\) members of \(\mathcal{F}\) have non-empty intersection, and it is called maximal \(k\)-wise intersecting if no family strictly containing \(\mathcal{F}\) satisfies this condition. We show that for each \(k\geq 2\) there is a maximal \(k\)-wise intersecting family of size \(O(2^{n/(k-1)})\). Up to a constant factor, this matches the best known lower bound, and answers an old question of Erdős and Kleitman, recently studied by Hendrey, Lund, Tompkins, and Tran.

Mathematics Subject Classifications: 05D05

Keywords: Intersecting family, saturation, set system