Let us pick one random vertex of the graph and assume it is a subset S with size k.
According to the question another vertices would be “adjacent if and only if the corresponding sets intersect in exactly two elements”.
Degree of this vertex = Other subsets that has exactly 2 elements in common with S.
For number of adjacent vertices, first we pick 2 elements that are common in both = kC2
Now we cannot take any other element from the rest of the k-2 elements of S1 because then there will be three or more elements common, but we want exactly two.
There are n-k elements remaining, for each of them we can either leave it or take it,
degree(S, k) = (kC2) * 2⁽ⁿ ⁻ ᵏ⁾
Now we have to find the value of k for which degree(S, k) will be maximum,
The maximum value of degree(S,k) = 3 * 2⁽ⁿ ⁻ ³⁾
[This value appears when k=3 and k=4, and it depends only on k as we can take 2ⁿ common from each term of degree(S,k) ]