4. At the party, 10 boys and 10 girls gathered. Each boy likes a different positive number of girls. Each girl likes a different positive number of boys. Determine the largest integer n with the following property: It is always possible to form n disjoint pairs where both individuals like each other.

(Josef Tkadlec)
SOLUTION. We will show that the largest $n$ sought is equal to 1.

In the first part of the solution, we will show that one suitable pair can always be formed. According to the problem statement, the numbers of girls liked by individual boys are all the numbers from 1 to 10 (in some order). The same applies to the numbers of boys liked by individual girls. One boy likes all 10 girls, and one girl likes all 10 boys, so this boy and this girl form a suitable pair.

In the second part of the solution, we will provide an example of a party that meets the conditions of the problem, in which there do not exist two disjoint pairs with mutual attraction. For its construction, arrange the boys in the top row and the girls below them in the bottom row as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_04_17_cb8de399e949329b1759g-07.jpg?height=280&width=1054&top_left_y=1008&top_left_x=501)

Assume that what is marked with arrows for the sixth boy and the sixth girl from the left is true: Each boy likes exactly the girl directly below him and all the girls to the right of her, while each girl likes exactly the boy first from the left and all the boys to the right of the boy above her. Then, clearly, each boy likes a different number of girls and each girl likes a different number of boys, but in all (ten) pairs with mutual attraction, only one boy is represented - the first one from the left. Therefore, no two such pairs are disjoint.