Glock–Kühn–Lo–Osthus, Kelly, and Kwan–Sah–Sawhney–Simkin conjectures

The entry concerns conjectures on sparse-configuration avoidance and enumeration, including Pasch-free Steiner triple systems and Latin squares without $2\times 2$ subsquares. The reported advance addresses critical transition regimes rather than proving all conjectures uniformly.

Known results

• Kwan, Sah, Sawhney, and Simkin (2022) established existence of Latin squares with arbitrarily high girth and derived lower bounds for intercalate-free Latin squares.
• Delcourt, Henderson, Lesgourgues, and Postle (2025) proved the Glock--Kühn--Osthus conjecture in a minimum-degree range $\delta(G)\ge 0.82733n$.
• Jain and Pham (2022) obtained threshold results for Latin squares and Steiner triple systems in random $3$-uniform hypergraphs.

October 2026 critical-regime estimates

Kwan and Pham derive a variational formula for triangle-free probabilities at $p=c/\sqrt{n}$, identify a phase transition near $c\approx 4.341$, and estimate counts of Pasch-free Steiner triple systems and Latin squares without $2\times 2$ subsquares, up to factors of $\exp(o(n^2))$. The manuscript explicitly provides estimates rather than a uniform resolution and is unrefereed.

Current status (as of October 2026): Several ranges and enumeration estimates are established, while the combined conjecture entry remains unresolved; the newest critical-regime claims are unverified.