Every sufficiently large triangle-divisible graph with minimum degree at least (7+√21)/14 + epsilon has a triangle decomposition with arbitrarily large girth.
Proof of the High Girth Existence Conjecture via Refined Absorption
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We prove the High Girth Existence Conjecture - the common generalization of the Existence Conjecture for Combinatorial Designs originating from the 1800s and Erd\H{o}s' Conjecture from 1973 on the Existence of High Girth Steiner Triple Systems.
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Erd\H{o}s meets Nash-Williams
Every sufficiently large triangle-divisible graph with minimum degree at least (7+√21)/14 + epsilon has a triangle decomposition with arbitrarily large girth.