Exercise 16, § 5

Let E be a finite set with $n$ elements, let $\mathfrak{D}$ be a non-empty subset of $\mathfrak{P}(E)$, and let $\lambda$, $k$, $l$ be three integers $\geqslant 1$ with the following properties : (i) if A, B are distinct elements of $\mathfrak{D}$, then Card $(A \cap B) = \lambda$; (ii) for each $A \in \mathfrak{D}$, Card $(A) \leqslant k$; (iii) for each $x \in E$ the number of elements of $\mathfrak{D}$ to which $x$ belongs is equal to $l$. Show that

and that if $n(\lambda - 1) = k(k - 1)$ then $\lambda = k$ and Card $(\mathfrak{D}) = n$. (Given $a \in E$, let $\mathfrak{L}$ be the set of all $A - \{a\}$ where $A \in \mathfrak{D}$ and $a \in A$, and let $\mathfrak{E}$ be the set of all $A \in \mathfrak{D}$ such that $a \notin A$. Apply the results of Exercise 15 to $\mathfrak{L}$ and $\mathfrak{E}$.)