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Matthew Kwan, Ashwin Sah, Mehtaab Sawhney and Michael Simkin prove, as Theorem 1.1 of High-girth Steiner triple systems (card), that for every $g$ there is $N(g)$ such that every $N\geq N(g)$ with $N\equiv1,3\pmod 6$ carries a Steiner triple system of order $N$ containing no $(j,j-2)$-configuration for any $4\leq j\leq g$, where a $(j,j-2)$-configuration is a set of $j-2$ triples spanning at most $j$ vertices. In the wording of Problem 207, a collection of $\ell$ triples spanning at most $\ell+2$ vertices is an $(\ell+2,\ell)$-configuration, so the theorem with $g$ replaced by $g+2$ says that any $\ell$ triples of the system span at least $\ell+3$ vertices for $2\leq\ell\leq g$; the cases $\ell=2,3$ hold in every Steiner triple system, as the paper notes, because two triples share at most one vertex and every $(5,3)$-configuration contains a $(4,2)$-configuration. The system is built as a triangle decomposition of $K_N$ by iterative absorption, with a high-girth triple process for the approximate decomposition and a sparse absorbing structure for the leftover; the paper names the difficulty that sparseness is not preserved under unions of partial systems and the constraint focusing it causes, and answers them by running the high-girth triple process first, so that the absorption works on a sparse leftover, and by analyzing the inherited forbidden configurations retrospectively through its weight systems rather than tracking them step by step. Erdős posed the question in his Rome 1973 problem paper (card, printed p. 9), asking whether for every $n>n_0(k)$ there is a Steiner system with no $G^{(3)}(r;r-2)$ for $3<r\leq k$, and reporting that Doyen could do this for $k=6$ and infinitely many $n$. Before this theorem only the $4$-sparse case was known for all large admissible orders, with partial results for $r=5$ and $r=6$ and no $7$-sparse system known, as the paper's survey of previous work states.

Acceptance. Refereed: Ann. of Math. (2) 200 (2024), no. 3, 1059–1156, after its first posting as arXiv:2201.04554 on 2022-01-12. Reviewed: Thomas Bloom, the site's curator, marks the problem proved and credits the proof to Kwan, Sah, Sawhney and Simkin [KSSS22b]. The formal-conjectures catalog states the problem, in a file added 2026-10-07 whose proof is left as sorry, and no Lean proof of the theorem is recorded, so the page lists no formalized evidence. The library card summarizes the paper's statements from its text and does not verify the proof; that reading is not acceptance evidence.