One-Factorizations

Suppose a balanced incomplete block design on v treatments has block size 2. When we say the elements x and y occur together in λ blocks, this is precisely the same as saying that {x, y} occurs as a block λ times. So the blocks are the 1

2 v(v − 1) unordered pairs of treatments, taken λ times each,

and the parameters are

(v, 12λv(v − 1), λ(v − 1), 2, λ). (10.1)

We have the trivial result

THEOREM 10.1

For every v and λ there is a unique balanced incomplete block design with k = 2, and its parameters are given by (10.1).