The Number of Triple Systems Without Even Cycles

Abstract

For k ⩾ 4, a loose k-cycle Ck is a hypergraph with distinct edges e1, e2, …, ek 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 ⩾ 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no Ck is at most $$2^{cn^2}$$ . An easy construction shows that the exponent is sharp in order of magnitude. 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.