The intersection numbers of KTSs with a common parallel class

In this paper, the intersection numbers of KTSs with a common parallel class are investigated. Denote by J 1 [ u ] the set of all integers k such that there is a pair of KTS( 3 u )s with a common parallel class intersecting in k + u triples, u of them being the triples of the common parallel class. It has been established that J 1 [ u ] = { 0 , 1 , … , 3 ( u 2 ) − 6 , 3 ( u 2 ) − 4 , 3 ( u 2 ) } for any odd integer u ≥ 7 and u ≠ 9 , 11 , 13 , 17 , 19 . For u = 9 , 11 , 13 , 17 , 19 , there are 11 cases that are left undecided.