Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

For n >= 3 and r = r(n) >= 3, let k = k(n) = (k(1), ..., k(n)) be a sequence of non-negative integers with sum M(k) = Sigma(n)(j=1) K-j. We assume that M(k) is divisible by r for infinitely many values of n, and restrict our attention to these values. Let X = X(n) be a simple r-uniform hypergraph on the vertex set V = {v(1), v(2), ..., v(n)} with t edges. We denote by H-r(k) the set of all simple r-uniform hypergraphs on the vertex set V with degree sequence k, and let H-r(k, X) be the set of all hypergraphs in H-r(k) which contain no edge of X. We give an asymptotic enumeration formula for the size of H-r(k, X). This formula holds when r(4)k(max)(3) = o(M(k)), t k(max)(3) = o(M(k)(2)) and r t k(max)(4) = o(M(k)(3)). Our proof involves the switching method. As a corollary, we obtain an asymptotic formula for the number of hypergraphs in H-r(k) which contain every edge of X. We apply this result to find asymptotic expressions for the expected number of perfect matchings and loose Hamilton cycles in a random hypergraph in H-r(k ) in the regular case. (C) 2018 Elsevier Ltd. All rights reserved.