A new result on Sylvester’s problem

Sylvester’s problem, i.e., the problem of the existence of large sets of Kirkman triple systems (LKTSs), is one of the most challenging problems in combinatorial design theory. Most recursive constructions for LKTSs impose resolvable restrictions on the ingredient designs, making the recursions difficult to apply. In this note we provide necessary small designs by using spherical geometries and a direct construction. Finally, we prove the existence of LKTS ( 12 × 7 n − 3 ) for all positive integers n . A new result on overlarge sets of Kirkman triple systems is also produced.