On the number of blocks in a generalized Steiner system

摘要

We consider t-designs with lambda=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. For t=2, this inequality is the well known De Bruijn-Erdos inequality. For t > 2 it has the same order of magnitude as the Wilson-Petrenjuk inequality for Steiner systems with constant block size. The point of this note is that the inequality is very easy to derive and does not seem to be known. A stronger inequality was derived in 1969 by Woodall (J. London Math. Soc. (2) 1, 509-519), but it requires Lagrange multipliers in the proof. (C) 1997 Academic Press.