Basic structures
Generating functions
Proof techniques
Combinatorial identities
Polytopes
Combinatorial design is a generic term for combinatorial structures described by families of finite sets satisfying some symmetries or other combinatorial properties of mutual arrangement. For example, the block designs generically describe a set with family of subsets satisfying some combinatorial properties.
A concrete structures of this kind are block -designs: if is an integer a -design is a set with a family of -element subsets of (called blocks) such that every appears in exactly blocks, and every -element subset appears in exactly blocks. One also says -design if is the cardinality of . The number of blocks and are determined by the other data. The applications include algebraic codes, finite geometries, algorithm design etc.
(Non)existence of combinatorial designs with specific properties often has profound consequences on classification of various other mathematical structures (not necessarily finite ones); in particular lattices, finite geometries, finite groups etc.
Related Lab items: binary linear code, synthetic projective geometry, Joyal species, matroid, building, incidence geometry
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.
Last revised on November 18, 2018 at 15:46:12. See the history of this page for a list of all contributions to it.