TITLE: Bound on cardinality of a union
QUESTION [6 upvotes]: Suppose I have n finite sets A1 through An contained in some fixed set S, and I am given non-negative integers N and N1 through Nn such that each Ai has cardinality N, and each k-tuple intersection has cardinality less than or equal to Nk.
Can I use this to construct a good lower bound on the cardinality of the union of the Ai?

REPLY [9 votes]: I don't know your reason for asking this question, so it's unlikely that what I'm about to write will be helpful. Nevertheless, there's an easy method I like a lot for deducing a lower bound just from the knowledge that N_2 is small. It may be contained in what has been said above -- I haven't checked.
The idea is to think of each set A_ i as a 01-valued function on a set of size M. We then look at the ell_ 2 norm of sum_i A_i. The square of the ell_ 2 norm is   sum_ {i,j}|A_ i cap A_ j|,   which by assumption is at most nN + n(n-1)N_ 2. But we also have a lower bound: the ell_ 1 norm of the sum is nN, from which it follows that the square of the ell_ 2 norm is at least (nN)^2/M.
Putting these two bits of information together gives a lower bound for M of nN/(1+(n-1)N_ 2/N). So, for example, if N_ 2 = cN for some smallish positive constant c, then the lower bound is roughly N/c.