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.
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).
@cite_38 @cite_8
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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. We study branching random walks in random environment on the d- dimensional square lattice, d 1. In this model, the environment has nite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, and the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process. 1. Introduction and results We start with an informal description of the model we study in this paper. Particles live in Z d and evolve in discrete time. At each time, every particle is substituted by (possibly more than one) ospring which are placed in neighboring sites, independently of the other particles. The rules of ospring generation depend only on the location of the particle. The collection of those rules (so-called the environment) is itself random, it is chosen randomly before starting the process, and then it is kept xed during all the subsequent evolution of the particle system.