Given two r-uniform hypergraphs G and H the Tur\'an number ex(G,H) is the maximum number of edges in an H-free subgraph of G. We study the
typical value of ex(G,H) when G=Gn,p(r)β, the Erd\H{o}s-R\'enyi
random r-uniform hypergraph, and H=C2β(r)β, the r-uniform linear
cycle of length 2β. The case of graphs (r=2) is a longstanding open
problem that has been investigated by many researchers. We determine the order
of magnitude of ex(Gn,p(r)β,C2β(r)β) for all
rβ₯4 and all ββ₯2 up to polylogarithmic factors for all values of
p=p(n).
Our proof is based on the container method and uses a balanced
supersaturation result for linear even cycles which improves upon previous such
results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan.Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:2007.1032