On the Sizes of k-edge-maximal r-uniform Hypergraphs

Let H = (V, E) be a hypergraph, where V is set of vertices and E non-empty subsets called edges. If all edges have the same cardinality r, then an r-uniform hypergraph; if consists r-subsets V, complete denoted by $$K_n^r$$ , n |V|. A hypergraph H′ (V′, E′) subhypergraph V′ ⊆ E′ E. The edge-connectivity minimum edge F such that − not connected, F). An k-edge-maximal every has at most k, but for any $$e \in \left({K_n^r} \right)\backslash E\left(H \right),\,H + e$$ contains least one with k 1. r integers ≥ 2 2, let t t(k, r) largest integer $$\left({\matrix{{t - 1} \cr {r \cr}} \right) \le k$$ . That is, satisfying < \left({\matrix{t \right)$$ We prove $$n \left| {V\left(H \right)} \right| \ge t$$ (i) $$\left| {E\left(H \left({n t} \right)k$$ this bound best possible. (ii) \right)k \left({\left({t \right)\left\lfloor {{n \over t}} \right\rfloor $$