What question did this study set out to answer?

To establish conditions under which a binomial random q-uniform hypergraph contains an (n,q,2)-Steiner system.

June 20, 2026Open Access

Thresholds for ( n , q , 2)-Steiner systems via refined absorption

MDMichelle DelcourtToronto Metropolitan University TKTom KellyGeorgia Institute of Technology LPLuke PostleUniversity of Waterloo

Key Points

To establish conditions under which a binomial random q-uniform hypergraph contains an (n,q,2)-Steiner system.
Proving asymptotic properties of binomial random q-uniform hypergraphs.
Analyzing the necessary divisibility conditions for n.
For p ≥ n^-(q-6)/2, the hypergraph G^(q)(n,p) almost surely contains an (n,q,2)-Steiner system if n meets divisibility conditions.

Abstract

Abstract We prove that if p greater than or equals n Superscript minus left parenthesis q minus 6 right parenthesis divided by 2 p ≥ n − (q − 6) / 2 p n^- (q-6) /2, then asymptotically almost surely the binomial random q -uniform hypergraph script upper G Superscript left parenthesis q right parenthesis Baseline left parenthesis n comma p right parenthesis G (q) (n, p) G^ (q) (n, p) contains an (n, q, 2) -Steiner system, provided n satisfies the necessary divisibility conditions.

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