Weakly union-free maximum packings (Q1306740)

The author improves the known results on the largest number of 3-subsets of an \(n\)-set for which no four distinct sets \(A\), \(B\), \(C\), \(D\) satisfy \(A\cup B=C\cup D\) by showing that the upper bound \(\lfloor n(n-1)/3 \rfloor\) is also met when \(n\equiv 2\pmod 3\), with few exceptions. (The cases \(n\equiv 0\), \(1\pmod 3\) were already settled.) The proof is by construction and both recursive techniques and transversal designs are used. Explicit packings are given for the small cases and a possible application to group testing and spread-spectrum communications is examined.