Balogh, József, Narayanan, Bhargav and Skokan, Jozef
(2019)
The number of hypergraphs without linear cycles.
Journal of Combinatorial Theory, Series B, 134.
pp. 309321.
ISSN 00958956
Abstract
The runiform linear kcycle C k r is the runiform hypergraph on k(r−1) vertices whose edges are sets of r consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to show that for any fixed pair of integers r,k≥3, the number of C k rfree runiform hypergraphs on n vertices is 2 Θ(n r−1) , thereby settling a conjecture due to Mubayi and Wang from 2017.
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