We have selected some proper subsets of a finite set $H$. We know that

a) any two different elements of the set are in exactly one selected subset;

b) for any selected subset $R$ and any element $p$ not in $R$, there are exactly $m$ selected subsets that contain $p$ and intersect $R$, and exactly $n$ selected subsets that contain $p$ and do not intersect $R$.

Express the number of elements of $H$ and the number of selected subsets in terms of $m$ and $n$. (H)
First, we show that any selected subset $R$ has exactly $m$ elements. Since $R$ is a proper subset, there exists an element $p$ not in it. By associating each $q \in R$ with the uniquely existing subset $S$ (as per (a)) that contains both $p$ and $q$, we obtain a one-to-one correspondence between the elements of $R$ and the selected subsets that contain $p$ and intersect with $R$. Therefore, every selected subset contains $m$ elements.

From condition (b), it follows that every element $p$ is contained in exactly $m+n$ selected subsets. Now, fix an element $r$ of $H$. Each element different from $r$ - by (a) - belongs to exactly one subset containing $r$. Since $r$ is contained in $m+n$ subsets, and each of these contains $m-1$ elements different from $r$, $H$ has a total of $h=1+(m+n)(m-1)$ elements.

Let the number of selected subsets be $k$. We count the pairs $(p, R)$ where $p$ is an element of $H$, $R$ is a selected subset, and $p \in R$. Since we have $h$ elements, and each is contained in $m+n$ subsets, the number of such pairs is $h(m+n)$. However, we can also count this by noting that each of the $k$ selected subsets contains $m$ elements, so the number of pairs is $k \cdot m$. The results of the two counting methods must be equal, so $k \cdot m = h(m+n)$, from which we get

$$
k=\frac{h(m+n)}{m}=\frac{[1+(m+n)(m-1)](m+n)}{m}
$$

Based on the work of Péter Frenkel (Fazekas M. Főv. Gyak. Gimn., III. o.t.)

Note. The problem did not include an investigation into whether there exist sets $H$ that satisfy the conditions. From the solution, it is clear - since $k$ must be an integer - that not every pair $(m; n)$ has a corresponding $H$. A necessary condition for existence is that $m \mid n(n-1)$; however, the question of existence is generally an open problem. The diagrams show two simple examples. The elements of $H$ are denoted by points $\left(P_{1}, P_{2}, \ldots\right)$, and the selected subsets are pairs of elements, which we represent by connecting the two points with a line. In the case of $m=n=3$, $H$ does exist (it has 13 elements, and 26 subsets must be selected), but specifying it is not as simple.
![](https://cdn.mathpix.com/cropped/2024_05_02_f964fe60951f29425b03g-1.jpg?height=1076&width=630&top_left_y=1128&top_left_x=736)