I suppose the answer to this question is easy but I don't have it.
Let $\mathcal{G}$ be the family of sets formed by the sets of points incident to each line of a finite projective plane with its order $n$ a prime power and built with vector space construction. For example for the Fano plane $\mathcal{G} = \{\{0,1,2\},\{0,3,4\}, \{0, 5, 6\}, \{1, 3, 5\}, \{2, 4, 5\}, \{2, 3, 6\}, \{1, 4, 6\}\}$. The number of elements of $\mathcal{G}$ is the number of points of the plane: $n^2+n+1$.
Let $\mathcal{F}$ be the closure of $\mathcal{G}$ under set union.
How can we show that all elements must have the same frequency, i.e. if $\mathcal{F}_a = \{X \in \mathcal{F} : a \in X\}$ and $\mathcal{F}_b = \{X \in \mathcal{F} : b \in X\}$ then $|\mathcal{F}_a| = |\mathcal{F}_b|$?
