  For $k \ge 4$, a loose $k$-cycle $C_k$ is a hypergraph with distinct edges
$e_1, e_2, \ldots, e_k$ such that consecutive edges (modulo $k$) intersect in
exactly one vertex and all other pairs of edges are disjoint. Our main result
is that for every even integer $k \ge 4$, there exists $c>0$ such that the
number of triple systems with vertex set $[n]$ containing no $C_{k}$ is at most
$2^{cn^2}$.
  An easy construction shows that the exponent is sharp in order of magnitude.
This may be viewed as a hypergraph extension of the work of Morris and Saxton,
who proved the analogous result for graphs which was a longstanding problem.
For $r$-uniform hypergraphs with $r>3$, we improve the trivial upper bound but
fall short of obtaining the order of magnitude in the exponent, which we
conjecture is $n^{r-1}$.
  Our proof method is different than that used for most recent results of a
similar flavor about enumerating discrete structures, since it does not use
hypergraph containers. One novel ingredient is the use of some (new)
quantitative estimates for an asymmetric version of the bipartite canonical
Ramsey theorem.
