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_40: The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.
  @cite_29: 
  @cite_7: Given a t-uniform hypergraph H on k vertices and an assignment of integers f(T) to thet -subsets T of a v-setX , v k+t, e give necessary and sufficient conditions for the existence of an assignment of integer multiplicities h(G) to those subhypergraphs G of the complete t-uniform hypergraph on v vertices that are isomorphic to H so that the sum of the integers h(G) over those G that contain T is f(T). Our main theorem is stated in terms of integral matrices. As a consequence of our main theorem, e also determine a diagonal form, and hence the p-rank for all primes p, for the incidence matrix of t-subsets versus subhypergraphs isomorphic to H.
summary: Another relaxation of the conjecture, which will play an important role in this paper, is obtained by considering integral designs', in which one assigns integers to the copies of @math in @math such that for every edge @math the sum of the integers assigned to the copies of @math containing @math is a constant independent of @math . Graver and Jurkat @cite_40 and Wilson @cite_29 showed that the divisibility conditions suffice for the existence of integral designs (this is used in @cite_29 to show the existence for large @math of integral designs with non-negative coefficients). Wilson @cite_7 also characterised the existence of integral @math -decompositions for any @math -graph @math .