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The existence of designs II
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The authors generalize the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes, and give approximate counting results for many structures whose existence was previously known.Abstract:
We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.read more
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The existence of designs via iterative absorption: hypergraph $F$-designs for arbitrary $F$
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References
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Book
Handbook of Combinatorial Designs
TL;DR: In this paper, the authors present a design theory of small-block designs of small order for the first time in the last half of the 20th century, starting from the design of the first block designs in the early 1950s.
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The existence of designs
TL;DR: The existence conjecture for combinatorial designs has been proved in this article for simplicial designs, answering a question of Steiner from 1853 and showing that the natural divisibility conditions are sufficient for clique decompositions of simplicial complexes that satisfy a certain pseudorandomness condition.
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An existence theory for pairwise balanced designs, III: Proof of the existence conjectures
TL;DR: The number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).
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An existence theory for pairwise balanced designs I. Composition theorems and morphisms
TL;DR: The general theorems presented here will be illustrated by examples and will be applied in the second part of this article, “An Existence Theory for Pairwise Balanced Designs, II.”