The number of triple systems without even cycles

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.