$\left(n,k,k-1\right)$-Steiner Systems in Random Hypergraphs

Abstract

Let $H$ be a random $k$-uniform $n$-vertex hypergraph where every $k$-tuple belongs to $H$ independently with probability $p$. We show that for some ${\epsilon }_k>0$, if $p\ge n^{-{\epsilon }_k}$, then asymptotically almost surely $H$ contains an $\left(n,k,k-1\right)$-Steiner System. Our main tool is Keevash's method of Randomized Algebraic Constructions.

22 pages