The packing of pairs by quadruples

Let X be a finite set of size ν, further let λ be a positive integer and let σ(4,λν) denote the maximum number of quadruples such that each pair of elements of X is contained in at most λ of them. The value of σ(4, 1;ν) has been determined by Brouwer (1979) for all v≥4. The value of σ(4,λν) has been determined by Billington, Stanton and Stinson (1984) for all ν≡0 (mod 3) and λ>1. In this paper we complete the determination of σ(4,λν) for all ν≥4 and λ>1.