Upper bound on cumulative power of system of limitedly intersecting subsets? We have a set $S$ of power $n$ and $k < n$ subsets $S_1, \ldots, S_k \subseteq S$  such that $|S_i \cap S_j| \le 1$ when $i \ne j$. Is there any nontrivial upper bound on total power of sets $S_1, \ldots, S_k$? In particular, is it true that
$$\sum_{i=1}^k |S_i| = O(n)$$

The Bonferroni inequalities (basically a truncated version of the inclusion–exclusion principle) can be brought to bear:
$$
\left|\bigcup_{i=1}^kS_i\right| \ge \sum_{i=1}^k |S_i| - \sum_{i<j}\left|S_i\cap S_j\right| \ge \sum_{i=1}^k|S_i| - \frac{k(k-1)}{2},
$$
hence
$$
\sum_{i=1}^k|S_i| \le \left|\bigcup_{i=1}^kS_i\right| + \frac{k(k-1)}{2} \le n + \frac{n(n-1)}{2},
$$
which is enough to bound your sum with $ O(n^2) $. I'm not sure you can do any better, although I wouldn't be surprised if your stricter bound were to be found to hold.
