Maximal Value of \(n\)

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Let \(A= \left \{a_1, \dots , a_n \right \}\) and \(S(A)\) be a set of some subsets \(D \subset A\) such that

  • each subset \(D \in S(A)\) contains at most \(n-1\) elements
  • each element of \(A\) belongs to exactly \(4\) distinct subsets
  • any unordered pair of distinct elements \(a_i, a_j \in A\) belongs to exactly one subset \(D\).

Determine the maximal possible value of \(n\).


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