My Solution: (I feel like I have a wrong understanding of sets which is why this seems so easy to me, someone correct me if im wrong, thanks)

Set E - S1, S2, … S50, X1, X2… Xn
Where S represents the subsets and X represent the elements part of set E.

Let’s say F has m amount of elements and it is a subset of E.
Therefore the maximum amount of elements in E is 2m-1 since each set has more than half the elements of set E.

Now lets think of a generic set Si, which is a subset of E. This set Si, must have more or equal to m amount of elements in order to qualify being a subset of E.
Since there are 2m-1 quantity of total elements in E, Si can have m-1 elements that are not part of set F. However it still needs 1 or more elements in order to have more or equal to m elements (which is a requirement to be a subset of E). As a result, at least 1 element will overlap with F.
This applies to all subset Si because all of them will require m or more elements to be a subset of E. This will result in overlapping with F due to the pigeonhole principle. Therefore it is possible to have a set F that contains an element in common with every set Si.

I may be wrong in my understanding of the question, please let me know my errors.