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A family of sets

Source: Bulgarian TST 2020 P3

January 23, 2021
combinatorics

Problem Statement

Let C\mathcal{C} be a family of subsets of A={1,2,…,100}A=\{1,2,\dots,100\} satisfying the following two conditions:
1) Every 9999 element subset of AA is in C.\mathcal{C}. 2) For any non empty subset C∈CC\in\mathcal{C} there is c∈Cc\in C such that C∖{c}∈C.C\setminus\{c\}\in \mathcal{C}.
What is the least possible value of ∣C∣|\mathcal{C}|?
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