Block design

Remarks

The family $\mathcal{B}$ may contain a block more than once. The design is simple if it has no repeated block. In a $2$-$(v, k, \lambda)$ design with $b$ blocks, each point lies in the same number $r$ of blocks, and $bk = vr$ and $r(k - 1) = \lambda (v - 1)$. Example: let $X = \set{1, \dots, 7}$ and let the blocks be $\set{1,2,3}$, $\set{1,4,5}$, $\set{1,6,7}$, $\set{2,4,6}$, $\set{2,5,7}$, $\set{3,4,7}$, $\set{3,5,6}$. Each of the $21$ pairs of points lies in exactly one of these blocks, so this is a $2$-$(7, 3, 1)$ design, the Fano plane. Here each point lies in $r = 3$ blocks, and $b = 7$. Non-example: on $X = \set{1, 2, 3, 4}$ the blocks $\set{1,2,3}$ and $\set{1,2,4}$ do not form a $2$-design, because the pair $\set{1,2}$ lies in two blocks and the pair $\set{3,4}$ lies in none. A $t$-$(v, k, 1)$ design is a Steiner system $S(t, k, v)$.