On λ-designs

A λ-design is a system of subsets S₁, S₂,…, S_n from an n-set S, n > 3, where |S_i ∩ S_j| = λ for i ≠ j, |S_j| = k_j > λ > 0, and not all k_j, are equal. Ryser 9] and Woodall 101 have shown that each element of S occurs either r₁, or r₂ times (r₁ ≠ r₂) among the sets S₁,…, S_n and r₁ +r₂ = n + 1. Here we: (i) mention most of what is currently known about λ-designs; (ii) provide simpler proofs of some known results; (iii) present several new general theorems; and (iv) apply our theorems and techniques to the calculation of all λ-designs for λ ? 5. In fact, this calculation has been done for all λ ?/ 9 and is available from the author.