Asymptotic packing and the random greedy algorithm

Let H be an r-uniform hypergraph satisfying deg(x) = D(1 + o(1)) for each vertex xϵ V(H) and deg(x, y) = o(D) for each pair of vertices x, y ϵ V(H), where D → infinity. Recently, J. Spencer [5] showed, using a branching process approach, that almost surely the random greedy algorithm finds a packing of size at least n/r(1 − o(1)) for this class of hypergraphs. In this paper, we show an alternative proof of this via “nibbles.” Further, let Tα be the number of edges that the random greedy algorithm has to consider before yielding a packing of size [n/r · (1 − α)]. We show that almost surely Tα ∼ (1/α)r−1 · n/r(r − 1) as α → 0+ holds.