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) = ∑n j=1 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 r 5k4max = o(M). A new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees is also presented, which holds when r4k3max = o(M). The proof focuses on the incidence matrix of a hypergraph, interpreted as a bipartite graph, and combines known enumeration results for bipartite graphs with a switching argument. ∗Research supported by the Australian Research Council. 1 a