Abstract A λ-design is a system of subsets S 1 , S 2 ,…, 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 r 1 , or r 2 times ( r 1 ≠ r 2 ) among the sets S 1 ,…, S n and r 1 + r 2 = 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.
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