Largest components in random hypergraphs

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n this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph. Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.
Original languageEnglish
Pages (from-to)741 - 762
JournalCombinatorics, Probability & Computing
Issue number5
Publication statusPublished - 2018

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