I think a found a satisfactory answer. Using [Corradi's Lemma][1] I can show ** for projective planes** ($\lambda=1$) I can show that the cardinality is at least $\frac{k^{2}s}{k+s-1}$. 


  [1]: http://books.google.co.il/books?id=OyrOOrTmlvwC&pg=PA24&lpg=PA24&dq=corradi%2527s+lemma&source=bl&ots=6GViNzUn4c&sig=Oqfl27w5eal8p1P405PnpAi1Oc4&hl=en&sa=X&ei=ZEiBUavOFMnWPPiJgZAD&ved=0CFoQ6AEwCA#v=onepage&q=corradi%2527s%2520lemma&f=false