The Number of Triple Systems Without Even Cycles

Abstract

For k 4, a loose k-cycle C k is a hypergraph with distinct edges e 1 , e 2 , . . . , 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 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.F as a (not necessarily induced) subgraph. Henceforth we will call G an F -free r-graph. Write Forb r (n, F ) for the set of F -free r-graphs with vertex set [n]. Since each subgraph of an F -free r-graph is also F -free, we trivially obtain |Forb r (n, F )| 2 ex r (n,F ) by taking subgraphs of an F -free r-graph on [n] with the maximum number of edges. On the other hand for fixed r and F , |Forb r (n, F )| i exr(n,F ) n r i = 2 O(exr(n,F ) log n) , so the issue at hand is the factor log n in the exponent. The work of Erdős-Kleitman-Rothschild [25] and Erdős-Frankl-Rödl [26] for graphs, and Nagle-Rödl-Schacht [45] for hypergraphs (see also [44] for the case r = 3) improves the upper bound above to obtain |Forb r (n, F )| = 2 ex r (n,F )+o(n r ) .Although much work has been done to improve the exponent above (see [1,6,7,8,31,34,48] for graphs and [10,11,21,47,13,50] for hypergraphs), this is a somewhat satisfactory state of affairs when ex r (n, F ) = Ω(n r ) or F is not r-partite.

The number of hypergraphs without linear cycles

“…for any fixed r, k ≥ 3. Since the lower bound in (1) is generally believed to be closer to the truth, Mubayi and Wang [12] made the natural conjecture that…”

“…Consequently, as in the case of graphs, it is important to understand the behaviour of prototypical examples of r-partite r-graphs. Here, following Mubayi and Wang [12], we shall investigate the enumeration problem for one such prototypical family of r-partite r-graphs, namely, the family of r-uniform linear (or loose) cycles.…”

On the number of linear hypergraphs of large girth

“…for every . Guided and motivated by this development on the extremal numbers of linear cycles, recently, Mubayi and Wang showed that for all even and improved the trivial upper bound in for . Inspired by Mubayi and Wang's method, Han and Kohayakawa subsequently improved the general upper bound to .…”

“…Guided and motivated by this development on the extremal numbers of linear cycles, recently, Mubayi and Wang showed that for all even and improved the trivial upper bound in for . Inspired by Mubayi and Wang's method, Han and Kohayakawa subsequently improved the general upper bound to . Very recently, Balogh, Narayanan, and Skokan provided a balanced supersaturation theorem for linear cycles and finally proved , for every , using the hypergraph container method .…”

Supersaturated Sparse Graphs and Hypergraphs

“…The idea of investigating Question 6 was suggested in a recent work of Mubyai and Wang [36]. They conjectured that Question 6 has an affirmative answer in the case when H is C (r) k , the r-uniform expansion 3 of C k , the (2-uniform) cycle of length k. Improving upon the result from [27,36], Balogh, Narayanan, and Skokan [7] have recently solved the conjecture of Wang and Mubayi. As immediate corollaries from Theorem 7 we reprove this result along with two related estimates for expansions of paths and complete bipartite graphs.…”