I have the following problem emerged. Let's say we have $l$ finite sets $A_1, A_2, \ldots, A_l$ with cardinality of $n_1, n_2, \ldots\, n_l$, respectively. We know that $| A_i \cap A_j | \le a_{ij}$ (ibvously, $a_{ij}=a_{ji}$).
What could be a lower bound on $\left|\bigcup_{k=1}^l A_k \right|$?
The bound I would be satisfied with should be easy (polynomial) to compute and thus could be rough (but better than just maximum of $n_i$).
Some notes
From inclusion-exclusion principle we know exact expresion for that: $$ \left|\bigcup_{k=1}^l A_k\right| = \sum_{k=1}^l (-1)^{k+1} \left( \sum_{1 \le i_1 < i_2 < \ldots < i_k \le l} \left| A_{i_1} \cap A_{i_2} \cap \ldots \cap A_{i_k} \right| \right) $$ but it is an exponintioal computation and we don't know all the intersections anyway.
In the real problem I am facing,there are the following values $$ n_i = w_i \binom{n-w_i}{k-1}, \\ a_{ij} = \delta \binom{n-w_i-w_j+\delta}{k-1}, \\ w_i, n, k, \delta \mbox{ are some natural numbers.} $$ But it's getting more messy to work with these expressions.
UPD. The question could be re-phrased like the following.
Find an easy computable function $\beta(l; a_{ij})$ such that for every $l$, $A_j$ and $a_{ij} = a_{ji}$ it holds $$ \left|\bigcup_{k=1}^l A_k\right| \ge \beta(l; a_{ij}) $$