A smoother notion of spread hypergraphs

Alweiss, Lovett, Wu, and Zhang introduced $q$ -spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$ -spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$ -spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph $G_{n,p}$ . In this paper, we give a common generalization of the original notion of $q$ -spread hypergraphs and the variant used by Kahn, Narayanan, and Park.