Independent Sets in Steiner Triple Systems

Definitions and mathematical statement

A Steiner triple system on a finite vertex set $V$ is a family $\mathcal{T}\subseteq\binom{V}{3}$ such that every two-element subset of $V$ belongs to exactly one member of $\mathcal{T}$. A set $I\subseteq V$ is independent if no member of $\mathcal{T}$ is contained in $I$. Define \[ \alpha(V,\mathcal{T})=\max\{|I|:I\subseteq V,\ \text{no }T\in\mathcal{T}\text{ satisfies }T\subseteq I\}. \] Here $\log$ is interpreted as the natural logarithm.

The proposed statement is: for every $\varepsilon\in(0,1)$, there exists an integer $N$ such that, for every integer $n\geq N$ with $n\equiv1$ or $3\pmod 6$, every Steiner triple system $(V,\mathcal{T})$ with $|V|=n$ satisfies \[ \alpha(V,\mathcal{T})\geq(1-\varepsilon)\sqrt{3n\log n}. \] Thus the source's asymptotic lower bound $(1+o(1))\sqrt{3n\log n}$ is understood uniformly over all Steiner triple systems of each admissible order.

The surrounding source passage also asks about an unspecified expected value and a bound apparently involving $d n^{1-r}\mathbb{E}(|I|^r)$. It does not define the random object, probability distribution, parameters, or expected quantity being bounded. Consequently, that associated shadow-moment question cannot be stated precisely from this record. The passage attributes the displayed Steiner triple system conjecture to reference [10] and the preceding question to [17], but neither numbered reference is identified in the supplied excerpt.

Short English statement

Does every sufficiently large Steiner triple system have a vertex subset containing no triple whose size is at least $(1-\varepsilon)\sqrt{3n\log n}$, for every fixed $\varepsilon>0$? The threshold must work for all systems of the same order. The supplied record labels this conjecture open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30006550$, title "Independent Sets in Steiner Triple Systems", source problem code catalogue entry.
• Supplied report citation: Combinatorics (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/1, pp. 5--79. https://doi.org/10.4171/owr/2026/1. The record describes the report as documenting workshop discussions and open problems. Its excerpt attributes the conjecture to [10] and the preceding expectation question to [17]; the corresponding bibliographic entries are not supplied.
• Supplied literature source: Kostochka--Mubayi--Verstraete, On independent sets in hypergraphs. https://arxiv.org/abs/1106.3098.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the proposed sharp asymptotic lower bound and associated shadow-moment estimate remain conjectural beyond known constant-factor bounds. This is supplied dated curation derived from the Oberwolfach report, not an independently verified literature assessment.
• The registry assigns statement_status: corrected_verified and states that the official report was checked, the question isolated, and its notation repaired. These are supplied verification claims; no source visit or independent verification is asserted here. No individual proposer is named in the record.