On $r$-uniform hypergraphs with circumference less than $r$

We show that for each $k\geq 4$ and $n>r\geq k+1$, every $n$-vertex $r$-uniform hypergraph with no Berge cycle of length at least $k$ has at most $\frac{(k-1)(n-1)}{r}$ edges. The bound is exact, and we describe the extremal hypergraphs. This implies and slightly refines the theorem of Gy\H{o}ri, Katona and Lemons that for $n>r\geq k\geq 3$, every $n$-vertex $r$-uniform hypergraph with no Berge path of length $k$ has at most $\frac{(k-1)n}{r+1}$ edges. To obtain the bounds, we study bipartite graphs with no cycles of length at least $2k$, and then translate the results into the language of multi-hypergraphs.