On the sizes of \((k, l)\)-edge-maximal \(r\)-uniform hypergraphs (Q2107753)

An \(r\)-uniform hypergraph \(H = (V, E)\) where \(|V(H)|=n\) is said to be \((k, l)\)-edge-maximal if every subhypergraph \(H^\prime\) of \(H\) with \(|V (H^\prime)| \geq l\) has edge-connectivity at most \(k\), but for any edge \(e \in E(K_{n}^{r}) \backslash E(H), H + e\) contains at least one subhypergraph \(H^{\prime\prime}\) with \(|V (H^{\prime\prime})| \geq l\) and edge-connectivity at least \(k+1\). In this paper, the authors obtain lower and upper bounds of the sizes of \((k, l)\)-edge-maximal \(r\)-hypergraphs of order \(n\) and show that these bounds are best possible.