Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th