Problem 207

../

claims/: The 1 claim page of Problem 207, one per claimant's result; the problem's standing derives from them.

---

Statement. For any $g\geq 2$, if $n$ is sufficiently large and $\equiv 1,3\pmod{6}$ then there exists a 3-uniform hypergraph on $n$ vertices such that every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and for any $2\leq j\leq g$ any collection of $j$ edges contains at least $j+3$ vertices.

Status. The site labels the problem proved, crediting Kwan, Sah, Sawhney and Simkin [KSSS22b]. The accepted claim is Steiner triple systems of arbitrarily high girth.

Source. erdosproblems.com/207, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #207, https://www.erdosproblems.com/207.

References.

• [Er76] Erdős, P., Problems and results in combinatorial analysis. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo II (1976), 3-17; the question is stated on printed p. 9. Library home: erdos_1976_problems_results_combinatorial_analysis.
• [KSSS22b] Kwan, M. and Sah, A. and Sawhney, M. and Simkin, M., High-girth Steiner triple systems. arXiv:2201.04554 (2022); Ann. of Math. (2) 200 (2024), no. 3, 1059-1156. Library home: kwan_2022_high_girth_steiner_triple_systems.

Formalization. Statement in formal-conjectures, added 2026-10-07 and marked solved there, with its proof left as sorry and no formal proof linked; no Lean proof of the theorem is recorded.

Current assessment

The site's formulation asks, for every $g\geq2$, for Steiner triple systems of every large admissible order in which any $j$ triples span at least $j+3$ vertices for $2\leq j\leq g$. The answer is yes: Kwan, Sah, Sawhney and Simkin prove it for every $g$, refereed in Ann. of Math. and credited by the site's curator; the problem's standing derives from that accepted claim. The threshold $N(g)$ is not quantified by the theorem and is not assessed here. Erdős's 1973 question [Er76] is the same statement in the language of $G^{(3)}(r;r-2)$ configurations, $r$ vertices spanned by $r-2$ triples, with $r=j+2$ and $k=g+2$; the cases $j=2,3$ ($r=4,5$) hold in every Steiner triple system.

Search scope, 2026-10-07: the site's page and discussion thread (no comments and no proof claims), the community database (teorth/erdosproblems), the formal-conjectures catalog (a statement file added 2026-10-07, without a proof) and Crossref. No other claim on the problem was found, and no Lean proof of the theorem is recorded anywhere.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• bohman_2019_large_girth_approximate_steiner_triple_systems
• bohman_2019_large_girth_approximate_steiner_triple_systems / theorem_1_3
• glock_2020_conjecture_erdos_locally_sparse_steiner_triple
• glock_2020_conjecture_erdos_locally_sparse_steiner_triple / conjecture_1_1
• glock_2020_conjecture_erdos_locally_sparse_steiner_triple / conjecture_7_2
• glock_2020_conjecture_erdos_locally_sparse_steiner_triple / theorem_1_2
• glock_2020_conjecture_erdos_locally_sparse_steiner_triple / theorem_1_3
• glock_2020_conjecture_erdos_locally_sparse_steiner_triple / theorem_4_4
• kwan_2022_high_girth_steiner_triple_systems
• kwan_2022_high_girth_steiner_triple_systems / theorem_1_1
• kwan_2022_high_girth_steiner_triple_systems / theorem_1_3