The triangle intersection numbers of a pair of disjoint s

In this paper the triangle intersection problem for a pair of disjoint S ( 2 , 4 , v ) s is investigated. Let J T ∗ ( v ) denote the set of all integers s such that there exists a pair of disjoint S ( 2 , 4 , v ) s intersecting in s triangles. Let b v = v ( v − 1 ) / 12 . We establish that for any positive integer v ≡ 1 , 4 ( mod 12 ) , v ≥ 16 and v ≠ 25 , 28 , 37 , J T ∗ ( v ) = [ 0 , b v ] .