A lower bound for the number of blocks of designs with odd

The aim of this paper is to give a new combinatorial proof of Fisher’s inequality and to prove that if t is odd, t > 1 , ε > 0 and b is the number of blocks of a t − ( v , k , λ ) design, then b ≧ ( 1 − ε ) ( λ 2 t − ( λ − 1 ) 2 t ) t 2 ( t − 1 ) − t 2 v t 2 for v ≧ v 0 .