Thresholds for ( <i>n</i> , <i>q</i> , 2)-Steiner systems via refined absorption

Abstract We prove that if p greater than or equals n Superscript minus left parenthesis q minus 6 right parenthesis divided by 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> $p \geq n^{-(q-6)/2}$ , 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msup> <mml:mrow> <mml:mi mathvariant="script" class="MJX-tex-caligraphic">G</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $\mathcal{G}^{(q)}(n,p)$ contains an ( n , q , 2)-Steiner system, provided n satisfies the necessary divisibility conditions.