No abstract

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 = (k 1 , . . . , k n ) be a vector of nonnegative integers, where each k j = k j (n) may depend on n. Let M = M (n) = n j=1 k j 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 k max satisfies r 4 k 4 max (k max + 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.

The problem of constructing asymptotic bounds for multiple packings in the space of q-ary sequences of length n is considered. For the zero rate, tightness of the expurgation bound is proved.