The existence of designs
  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. 
Peter Keevash
math.CO