Let $D$ be a $(v,k,\lambda)$-design (repeated blocks are allowed). I would like to get a lower bound on the cardinality of the union of $s$ blocks. A naive application of inclusion-exclusion gives $sk-\binom{k}{2}$ which is sometimes useful, but from the few examples I've worked out seems to be a severe underestimation of the true situation.

Has anyone treated this question before?

If it helps, we can progressively simplify to symmetric designs and then to finite projective planes (i.e. $\lambda=1$).