On The Set \(J_o(v)\) for Steiner Quadruple Systems

We prove that for every \(v \equiv 4, 8 \pmod{12}\) with \(v \geq 16\), there exists a pair of \(S(3,4,v)\)s having exactly \(k \in \{0, 1, \ldots, \lfloor \frac{v}{4} \rfloor \}\) pairwise disjoint blocks in common.