In the company, each employee has at least 50 acquaintances. It turned out that there are two employees who are familiar only through 9 handshakes (i.e. the shortest way of communication consists of 8 intermediate people). Prove that at least 200 employees work in this company.
My attempts of solution are straightforward. Consider a graph in which the vertices are people, and the edge denotes the familiarity of people. Each vertex has 50 edges and there are 2 vertices whose shortest distance is 9. We enumerate the path. Note that the vertices connected to the first vertex can not coincide with the vertices connected to a 5 or 6 vertex. Similarly, vertices connected to 5 or 6 vertices can not intersect with vertices connected to 10 vertex. Otherwise, the graph would have a path shorter than 9. Thus there are at least 150 vertices in the graph. However, I do not know how to prove about 200 vertices.
Thank you for any help or ideas!