On the Sizes of (k, l)-Edge-Maximal r-Uniform Hypergraphs

Let = ( ) be a hypergraph, where is a set of vertices and is a set of non-empty subsets of called edges. If all edges of have the same cardinality , then is an -uniform hypergraph; if consists of all -subsets of , then is a complete -uniform hypergraph, denoted by , where = | |. An -uniform hypergraph = ( ) is ( )-edge-maximal if every subhypergraph ′ of with | ( ′)| ≥ has edge-connectivity at most , but for any edge ) \ ), + contains at least one subhypergraph ′′ with | ( ′′)| ≥ and edge-connectivity at least +1. In this paper, we obtain the lower bounds and the upper bounds of the sizes of ( )-edge-maximal hypergraphs. Furthermore, we show that these bounds are best possible.