Let: To be a balanced incomplete block design it is required that: Fisher's inequality states simply that Let the incidence matrix M be a v × b matrix defined so that Mi,j is 1 if element i is in block j and 0 otherwise.
A pairwise balanced design (or PBD) is a set X together with a family of non-empty subsets of X (which need not have the same size and may contain repeats) such that every pair of distinct elements of X is contained in exactly λ (a positive integer) subsets.
The size of X is v and the number of subsets in the family (counted with multiplicity) is b. Theorem: For any non-trivial PBD, v ≤ b.
[1] This result also generalizes the Erdős–De Bruijn theorem: For a PBD with λ = 1 having no blocks of size 1 or size v, v ≤ b, with equality if and only if the PBD is a projective plane or a near-pencil (meaning that exactly n − 1 of the points are collinear).
[2] In another direction, Ray-Chaudhuri and Wilson proved in 1975 that in a 2s-(v, k, λ) design, the number of blocks is at least