The Existence of BIB Designs

Given any positive integers k. 3 and ., let c(k, .) denote the smallest integer such that v.B(k, .) for every integer v.c(k, .) that satisfies the congruences .v(v. 1) . 0(mod k(k. 1)) and .(v. 1) . 0(mod k. 1). In this article we make an improvement on the bound of c(k, .) provided by Chang in [4] and prove that c(k,.).exp{k3k6}. In particular, c(k,1).exp{kk2}.