The Minimum Number Of Subsets

Find the last three digits of the smallest positive integer $n$ with the following property:

Let $\mathcal{S}$ be any set containing $n$ elements. Partition the $5$ element subsets of $\mathcal{S}$ into two partitions. Then, at least one of the partitions must contain $2014$ pairwise disjoint sets.

Details and assumptions

• Sets $A_1, A_2, \cdots , A_k$ are called pairwise disjoint if $|A_i \cap A_j| = 0$ for all $i \neq j.$

• This condition must hold for all sets $\mathcal{S}$ containing $n$ elements and all partitions of its $5$ element subsets.

• This problem is a generalization of an old USAMO problem.

• If the last three digits of $n$ are $012,$ enter $12$ as your answer.