Count Friends at a Party

Solution: Let the party correspond to a graph on $2025$ vertices connecting by edges as “friends”. The condition “every pair of two people has exactly one common friend” is the friendship condition.

The Friendship Theorem says any finite graph with that property is a windmill (friendship) graph: there is one central vertex joined to every other vertex, and the remaining vertices split into disjoint pairs so that each pair together with the center forms a triangle.

Because $2025=2\times 1012+1$, the graph is the windmill with $1012$ triangles glued at the center. Therefore:

• the maximum number of friends, the center person, most likely the host, is $2025−1=2024$;

• the minimum number of friends, any non-center person, is $2$, i.e. the center and their paired partner.