A lower bound for the number of blocks of t−(v,k,λ)t−(v,k,λ) designs with tt odd

The aim of this paper is to give a new combinatorial proof of Fisher’s inequality and to prove that if tt is odd, t>1t>1, ε>0ε>0 and bb is the number of blocks of a t−(v,k,λ)t−(v,k,λ) design, then b≧(1−ε)(λ2t−(λ−1)2t)t2(t−1)−t2vt2 for v≧v0v≧v0.