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Refined Absorption: A New Proof of the Existence Conjecture

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arxiv 2402.17855 v1 pith:FXUS7NVW submitted 2024-02-27 math.CO

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keywords existenceconjecturemethodabsorptioncombinatorialdesignsproofalternate
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abstract

The study of combinatorial designs has a rich history spanning nearly two centuries. In a recent breakthrough, the notorious Existence Conjecture for Combinatorial Designs dating back to the 1800s was proved in full by Keevash via the method of randomized algebraic constructions. Subsequently Glock, K\"{u}hn, Lo, and Osthus provided an alternate purely combinatorial proof of the Existence Conjecture via the method of iterative absorption. We introduce a novel method of refined absorption for designs; here as our first application of the method we provide a new alternate proof of the Existence Conjecture (assuming the existence of $K_q^r$-absorbers by Glock, K\"{u}hn, Lo, and Osthus).

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  1. Erd\H{o}s meets Nash-Williams

    math.CO 2025-07 conditional novelty 8.0 of 10

    Every sufficiently large triangle-divisible graph with minimum degree at least (7+√21)/14 + epsilon has a triangle decomposition with arbitrarily large girth.

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