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Felix Goldberg
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Lower bounds on cardinality of a union of blocks in a design

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. $\lambda=1$).

Felix Goldberg
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