On the Random Turán Number of Linear Cycles

ABSTRACT Given two ‐uniform hypergraphs and the Turán number is the maximum number of edges in an ‐free subgraph of . We study the typical value of when , the Erdös‐Rényi random ‐uniform hypergraph, and , the ‐uniform linear cycle of length . The case of graphs () is a longstanding open problem that has been investigated by many researchers. We determine the order of magnitude of for all and all up to polylogarithmic factors for all values of . 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.