Nearly Optimal Partial Steiner Systems

A partial Steiner system Sp(t, k, n) is a collection of k-subsets (i.e. subsets of size k) of n element set such that every t-subset is contained in at most one k-subset. To avoid trivial cases, we assume 2 ⩾ t < k < n. It is easy to see that the size of a partial Steiner system Sp(t, k, n) is at most (tn)/(kt). Confirming a conjecture of Erdos and Hanani [5], Rödl [19] proved that for fixed positive integers t and k, there is a partial Steiner system Sp(t, k, n) of size at least (1 − o(1)) ( nt)( kt) o(1) goes to 0 as n goes to infinity. Grable [7] found an explicit bound of the o(1) term above and it was improved by Kostochka and Rödl [15] to n−1/((kt)+o(1)−1). All proofs used Rödlʹs nibble method. We also use the method to improve the bound to O(( log nn) k − t(( kt)−1) ⩾ t < k — 2. (For t = k — 1, slightly better bounds were already known. See [2].) We believe that the nibble method gives no better bound up to a logarithmic factor.