Fisher's inequality

Remarks

Proof idea: let $r$ be the number of blocks through a point, and let $N$ be the $v \times b$ incidence matrix of points against blocks. Then $N N^{T} = (r - \lambda) I + \lambda J$, where $I$ is the identity matrix and $J$ is the all-ones matrix of size $v$. Its determinant is $rk (r - \lambda)^{v - 1}$. From $r(k - 1) = \lambda(v - 1)$ and $k < v$ one gets $r > \lambda$, so the determinant is not zero. So $N$ has rank $v$, and a $v \times b$ matrix of rank $v$ needs $b \ge v$. Example: the Fano plane is a $2$-$(7, 3, 1)$ design with $b = v = 7$, so the bound is sharp. The designs with $b = v$ are the symmetric designs; the Bruck–Ryser–Chowla theorem restricts their parameters. The hypothesis $k < v$ is needed. If $k = v$, every block is the whole set $X$, and $\lambda$ copies of $X$ form a design with $b = \lambda$ blocks, which can be less than $v$.

Going deeper

For a student

Fisher's inequality is a necessary condition for the existence of a design, beyond the divisibility conditions $bk = vr$ and $r(k-1) = \lambda(v-1)$. Its proof by the rank of the incidence matrix is a model of the linear algebra method in combinatorics. The equality case $b = v$ leads to symmetric designs, such as finite projective planes and the designs built from difference sets.