My book has a problem: Describe a group of five people, any two of whom have exactly one friend in common. Can you find a group of four people with this same property?
I found such a group of five people (the graph looks like two triangles with one common vertex), and I'm fairly sure there is no such 4 vertex graph. However, I'm having trouble finding a more general explanation for my solution. Is there any way to categorize which n allow such a graph, and how you could construct it?