The number of hypergraphs without linear cycles

The r-uniform linear k-cycle C k r is the r-uniform 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 r-free r-uniform hypergraphs on n vertices is 2 Θ(n r−1) , thereby settling a conjecture due to Mubayi and Wang from 2017

3-uniform hypergraphs and linear cycles

We continue the work of Gyárfás, Győri and Simonovits [Gyárfás, A., E. Győri and M. Simonovits, On 3-uniform hypergraphs without linear cycles. Journal of Combinatorics 7 (2016), 205–216], who proved that if a 3-uniform hypergraph H with n vertices has no linear cycles, then its independence number α≥[Formula presented]. The hypergraph consisting of vertex disjoint copies of complete hypergraphs K5 3 shows that equality can hold. They asked whether α can be improved if we exclude K5 3 as a subhypergraph and whether such a hypergraph is 2-colorable. We answer these questions affirmatively. Namely, we prove that if a 3-uniform linear-cycle-free hypergraph H, doesn't contain K5 3 as a subhypergraph, then it is 2-colorable. This result clearly implies that α≥⌈[Formula presented]⌉. We show that this bound is sharp. Gyárfás, Győri and Simonovits also proved that a linear-cycle-free 3-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free 3-uniform hypergraph has a vertex of degree at most n−2 when n≥10. © 2017 Elsevier B.V