Abstract
Ford≥ 2, letHd(n,p)denote a randomd-uniform hypergraph withnvertices in which each of the$\binom{n}{d}$possible edges is present with probabilityp=p(n)independently, and letHd(n,m)denote a uniformly distributedd-uniform hypergraph withnvertices andmedges. Let eitherH=Hd(n,m)orH=Hd(n,p), wherem/nand$\binom{n-1}{d-1}p$need to be bounded away from (d−1)−1and 0 respectively. We determine the asymptotic probability thatHis connected. This yields the asymptotic number of connectedd-uniform hypergraphs with given numbers of vertices and edges. We also derive a local limit theorem for the number of edges inHd(n,p), conditioned onHd(n,p)being connected.
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