Let D be a (v,k,λ)-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).