Steiner triple systems with disjoint or intersecting subsystems

The existence of incomplete Steiner triple systems of order v having holes of orders w and u meeting in z elements is examined, with emphasis on the disjoint (z = 0) and intersecting (z = 1) cases. When w greater than or equal to u and v = 2w + u - 2z, the elementary necessary conditions are shown to be sufficient for all values of z. Then for z is an element of (0, 1) and v "near" the minimum of 2w + u - 2z, the conditions are again shown to he sufficient. Consequences for larger orders are also discussed, in particular the proof that when one hole is at least three times as large as the other, the conditions are again sufficient. (C) 2000 John Wiley & Sons, Inc.