On the Number of Blocks in a Generalized Steiner System

We considert-designs withλ=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. Fort=2, this inequality is the well known De Bruijn–Erdős inequality. Fort>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.