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

TL;DR

This paper demonstrates that in a random hypergraph with sufficiently high probability, an $(n, k, k-1)$-Steiner System exists asymptotically almost surely, using advanced algebraic construction methods.

Contribution

It establishes the existence of Steiner systems in random hypergraphs under new probabilistic conditions, extending previous combinatorial existence results.

Findings

Steiner systems exist in random hypergraphs with probability p ≥ n^{- ext{epsilon}_k}

Asymptotic almost sure existence of Steiner systems under these conditions

Application of Keevash's Randomized Algebraic Constructions method

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 $\varepsilon_k > 0$, if $p \geq n^{-\varepsilon_k}$, 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.