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The existence of designs

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arxiv 1401.3665 v4 pith:45CT4U6C submitted 2014-01-15 math.CO

Peter Keevash This is my paper · ORCID
classification math.CO
keywords existencecliquedesignsansweringassumingcertaincombinatorialcomplexes
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We prove the existence conjecture for combinatorial designs, answering a question of Steiner from 1853. More generally, we show that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition. As a further generalisation, we obtain the same conclusion only assuming an extendability property and the existence of a robust fractional clique decomposition.

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Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  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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  3. Perfect Matchings in Random Sparsifications of Dense Hypergraphs

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  4. Linear Tur\'an Numbers of Uniform Hypertrees

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    For several r-uniform linear hypertrees with four edges, the maximum number of edges in a linear r-uniform hypergraph avoiding them is determined; the 4-uniform 4-edge path case is settled exactly.

  5. A remark on the $t$-intersecting Erd\H{o}s-Ko-Rado theorem

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  6. Delta-system method: a survey

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    A survey of the Delta-system (sunflower) method in extremal set theory, with proofs of key theorems and a broad literature review.

  7. Entropy methods in combinatorics

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    A selective survey of entropy methods in combinatorics, detailing randomized chain rules, Shearer's inequality, random homomorphisms, Pinsker-type arguments, the union-closed sets breakthrough, and entropy approaches ...

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