The determination of all λ-designs with λ=3

AbstractLet S1, …, Sn, n>1, be subsets of an n-set S where |Si|>λ≥1 and |Si∩Sj|=λ for i≠j. Then our configuration is either a symmetric block design, with possible degeneracies, or what Ryser [3] has called a λ-design. A λ-design has the remarkable property, established by Ryser [3], that each element of S occurs either r1 or r2 times among the sets Si, …, Sn and r1+r2=n+1. The 1-designs are completely known and so is the unique 2-design. The present paper establishes that there are exactly three 3-designs