Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For n≥ 3, let r= r(n)≥ 3 be an integer and let k = (k₁,…, kₙ) be a vector of nonnegative integers, where each kⱼ = kⱼ(n) may depend on n. Let M = M(n) = ∑ⱼ₌₁ⁿ kⱼ for all n≥ 3, and define the set I = \ n≥ 3 r(n) divides M(n)\. We assume that I is infinite, and perform asymptotics as n tends to infinity along I. Our main result is an asymptotic enumeration formula for linear r-uniform hypergraphs with degree sequence k. This formula holds whenever the maximum degree kₘₐₓ satisfies r⁴ kₘₐₓ⁴(kₘₐₓ + r) = o(M). Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.