Threshold for Steiner triple systems

2022/04/08 by Ashwin Sah, Mehtaab Sawhney, Sah, Ashwin +3 · 2 citations
Computer Science · Engineering · Mathematics · #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2204.03964

openalex publication_date 2022/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that with high probability \mathbbG⁽³⁾(n,n^-1+o(1)) contains a spanning Steiner triple system for n≡ 1,3\pmod6, establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.

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• Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory
• No-(k+1)-in-line problem for large constant k

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