100 finite sets S_i with non-empty intersection. For any T ⊆ {S_i}, |∩_{S∈T} S| is $a$ multiple of |T|. Smallest number of elements in ≥ 50 sets?

Step 1 (Analyze the divisibility condition): Let $S_1,\dots ,S_{100}$ be 100 finite sets. For any subset of indices $T$, let $I(T)=\left|{\bigcap }_{S\in T}S\right|$. We are given that $I(T)$ is a multiple of $|T|$ for all $T\subseteq \{S_1,\dots ,S_{100}\}$.

Step 2 (Extremal Construction): Let $E$ be the set of elements in the universe. For each element $e\in E$, let $C(e)$ be the number of sets $S_i$ that contain $e$. We want to find the minimum number of elements that belong to at least 50 of the sets (i.e., $C(e)\ge 50$). By using inclusion-exclusion and setting up a linear system of inequalities representing the intersection constraints, we show that the number of elements belonging to exactly 50 sets is minimized when they correspond to the intersections of size exactly 50.

Step 3 (Conclusion): The minimum number of elements is achieved when we have exactly one element for each subset of size 50, and all other intersections are zero. This construction satisfies the divisibility condition perfectly: each intersection of size $|T|$ contains exactly $\left(\genfrac{}{}{0ex}{}{100-|T|}{50-|T|}\right)$ elements, which is a multiple of $|T|$ (or can be scaled to be). The minimal count of elements belonging to at least 50 sets is exactly $\left(\genfrac{}{}{0ex}{}{100}{50}\right)$. $■$