Fisher's inequality

Fisher's inequality is a necessary condition for the existence of a balanced incomplete block design, that is, a system of subsets that satisfy certain prescribed conditions in combinatorial mathematics. Outlined by Ronald Fisher, a population geneticist and statistician, who was concerned with the design of experiments such as studying the differences among several different varieties of plants, under each of a number of different growing conditions, called blocks.

Let:


 * $v$ be the number of varieties of plants;
 * $b$ be the number of blocks.

To be a balanced incomplete block design it is required that:


 * $k$ different varieties are in each block, $1 ≤ k < v$; no variety occurs twice in any one block;
 * any two varieties occur together in exactly $λ$ blocks;
 * each variety occurs in exactly $r$ blocks.

Fisher's inequality states simply that



Proof
Let the incidence matrix $b ≥ v$ be a $M$ matrix defined so that $v × b$ is 1 if element $M_{i,j}$ is in block $i$ and 0 otherwise. Then $j$ is a $B = MM^{T}$ matrix such that $v × v$ and $B_{i,i} = r$ for $B_{i,j} = λ$. Since $i ≠ j$, $r ≠ λ$, so $det(B) ≠ 0$; on the other hand, $rank(B) = v$, so $rank(B) ≤ rank(M) ≤ b$.

Generalization
Fisher's inequality is valid for more general classes of designs. A pairwise balanced design (or PBD) is a set $v ≤ b$ 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 $X$ (a positive integer) subsets. The set $λ$ is allowed to be one of the subsets, and if all the subsets are copies of $X$, the PBD is called "trivial". The size of $X$ is $X$ and the number of subsets in the family (counted with multiplicity) is $v$.

Theorem: For any non-trivial PBD, $b$.

This result also generalizes the Erdős–De Bruijn theorem:

For a PBD with $v ≤ b$ having no blocks of size 1 or size $v$, $λ = 1$, with equality if and only if the PBD is a projective plane or a near-pencil (meaning that exactly $v ≤ b$ of the points are collinear).

In another direction, Ray-Chaudhuri and Wilson proved in 1975 that in a $n − 1$ design, the number of blocks is at least $$\binom{v}{s}$$.