Asymptotic linearity of binomial random hypergraphs via cluster expansion under graph-dependence

Abstract

Let integer n⩾3 and integer r=r(n)⩾3. Define the binomial random r-uniform hypergraph Hr(n,p) to be the r-uniform graph on the vertex set [n] such that each r-set is an edge independently with probability p. A hypergraph is linear if every pair of hyperedges intersects in at most one vertex. We study the probability of linearity of random hypergraphs Hr(n,p) via cluster expansion and give more precise asymptotics of the probability in question, improving the asymptotic probability of linearity obtained by McKay and Tian, in particular, when r=3 and p=o(n−7/5).