The existence of BIB designs (Q1569958)

The author improves his estimate [J. Comb. Des. 4, No. 2, 83-93 (1996; Zbl 0913.05017)] of the upper bound for the smallest integer \(c(k,\lambda)\) such that \(v\in B(k,\lambda)\) for every integer \(v\geq c(k,\lambda)\) that satisfies the congruences \(\lambda v(v- 1)\equiv 0\;(\text{mod }k(k- 1))\) and \(\lambda(v- 1)\equiv 0\;(\text{mod }k- 1)\) in a balanced incomplete block design (BIB design) \(B(k,\lambda; v)\). He proves that \(c(k,\lambda)\leq \exp\{k^{3k^6}\}\) and \(c(k,1)\leq \exp\{k^{k^2}\}\).