The determination of all λ-designs with λ=3

Abstract

Let S₁, …, S_n, n>1, be subsets of an n-set S where |S_i|>λ≥1 and |S_i∩S_j|=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 S_i, …, S_n 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.