Steiner triple systems with disjoint or intersecting subsystems

The existence of incomplete Steiner triple systems of order υ 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 <UEQN NOTAT="TEX" LOC="INLINE">$w \ge u$</UEQN> and <UEQN NOTAT="TEX" LOC="INLINE">$\upsilon=2w+u-2z$</UEQN>, the elementary necessary conditions are shown to be sufficient for all values of z. Then for <UEQN NOTAT="TEX" LOC="INLINE">$z\in\{0,1\}$</UEQN> and υ “near” the minimum of <UEQN NOTAT="TEX" LOC="INLINE">$2w+u-2\,z$</UEQN>, the conditions are again shown to be 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. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 58–77, 2000