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Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is aset of non-empty subsets of $V$ called edges. If all edges of $H$ have the samecardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all$r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by$K_n^r$, where $n=|V|$. A hypergraph $H'=(V',E')$ is called a subhypergraph of$H=(V,E)$ if $V'\subseteq V$ and $E'\subseteq E$. A $r$-uniform hypergraph$H=(V,E)$ is $k$-edge-maximal if every subhypergraph of $H$ hasedge-connectivity at most $k$, but for any edge $e\in E(K_n^r)\setminus E(H)$,$H+e$ contains at least one subhypergraph with edge-connectivity at least$k+1$.  Let $k$ and $r$ be integers with $k\geq2$ and $r\geq2$, and let $t=t(k,r)$ bethe largest integer such that $(^{t-1}_{r-1})\leq k$. That is, $t$ is theinteger satisfies $(^{t-1}_{r-1})\leq k<(^{t}_{r-1})$. We prove that if $H$ isa $r$-uniform $k$-edge-maximal hypergraph such that $n=|V(H)|\geq t$, then  ($i$) $|E(H)|\leq (^{t}_{r})+(n-t)k$, and this bound is best possible;  ($ii$) $|E(H)|\geq (n-1)k -((t-1)k-(^{t}_{r}))\lfloor\frac{n}{t}\rfloor$, andthis bound is best possible.  This extends former results in [8] and [6].

On the sizes of $k$-edge-maximal $r$-uniform hypergraphs