text: 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.
  @cite_38: We study several classes of arrays generalizing designs and orthogonal arrays. We use these to construct non-trivial t-designs without repeated blocks for all t.
  @cite_8: We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, @math -designs, and @math -wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.
summary: Even the existence of designs with @math and any non-trivial' @math was open before the breakthrough result of Teirlinck @cite_38 confirming this. An improved bound on @math and a probabilistic method (a local limit theorem for certain random walks in high dimensions) for constructing many other rigid combinatorial structures was recently given by Kuperberg, Lovett and Peled @cite_8 . Their result for designs is somewhat complementary to ours, in that they can allow the parameters @math and @math to grow with @math , whereas we require them to be (essentially) constant. They also obtain much more precise estimates than we do for the number of designs (within their range of parameters).