Problem

Source: 2020 Macedonian National Olympiad

Tags: combinatorics, set theory, national olympiad



Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met: (i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ; (ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$. Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.