I was curious whether there are any inequalities that relate the summation terms that appear in the formula for the inclusion exclusion principle (IEP).
More specifically, given a family $F$ of sets, if we denote the sum of the cardinalities of all k-wise intersections by $S_k$, then the IEP states that
$ \left| \bigcup_{i=1}^n A \right| = S_1 - S_2 + S_3 - ... $
where $S_k = \sum_{1 \leq i_1 < ... < i_k \leq n} \left| A_{i_1} \cap ... \cap A_{i_k} \right|$.
My questions are:
Are there any non-trivial inequalities relating the $S_i$ values for general (or possibly more restricted) set systems?
Are there any references related to this which you could point me to? I haven't really been able to find anything online.