Erdős' high-girth Steiner triple system conjecture

A (j,i)(j,i)-configuration in a family of sets is a collection of ii members spanning at most jj ground-set elements; the girth of a triangle packing is the smallest i≥2i\geq 2 for which it contains an (i+2,i)(i+2,i)-configuration. Erdős' conjecture. For every integer g≥3g\geq 3, every sufficiently large K3K_3-divisible complete graph admits a K3K_3-decomposition with girth at least gg. The source states that Kwan, Sah, Sawhney, and Simkin proved this conjecture in full, so it is solved.

References

Primary source

Luke Postle, “Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory”, arXiv:2510.19978 (2025).

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