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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. © 1996 John Wiley & Sons, Inc.

A partial Steiner (n, k, l)-system or briefly (n, k, l)-system is a pair (V, S), where V is an n-set and S is a collection of k-subsets of V, such that every l-subset of V is contained in at most one k-subset of S. A subset X ⊂ V is called independent if [X]k ∩ S = 0. The size of the largest independent set in S is denoted by α(S). Define The purpose of this note is to prove that for every k, l, k > l holds, where c, d are positive constants depending on k and l only.

Say a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function fr(n) = max{min{|V(H)|: H (G)1r}: |V(G)| = n}, where H (G)1r indicates that H is vertex Ramsey to G, that is, any vertex coloring of H with r colors admits a monochromatic induced copy of G. The method used provides a better upper bound for this function than was previously known. As a tool, we establish a theorem for projective planes.