Combinatorial design

Remarks

A $2$-$(v, k, \lambda)$ design is a balanced incomplete block design, and a $t$-$(v, k, 1)$ design is a Steiner system $S(t, k, v)$. Counting gives the number of blocks: $\abs{\mathcal{B}} = \lambda \binom{v}{t} / \binom{k}{t}$.

Example: $X = \set{1, \dots, 7}$ with blocks $\set{1,2,4}$, $\set{2,3,5}$, $\set{3,4,6}$, $\set{4,5,7}$, $\set{5,6,1}$, $\set{6,7,2}$, $\set{7,1,3}$ is a $2$-$(7, 3, 1)$ design, the Fano plane. The blocks are the translates of $\set{1, 2, 4}$ modulo $7$, whose pairwise differences $\pm 1, \pm 2, \pm 3$ cover each nonzero residue exactly once, so every pair of points lies in exactly one block.

Non-example: $X = \set{1, 2, 3, 4}$ with blocks $\set{1, 2}$ and $\set{3, 4}$ is not a $2$-$(4, 2, 1)$ design, because the pair $\set{1, 3}$ lies in no block.