A finite projective plane is a hypergraph in which 1. Any two edges shere exactly one node 2. There is exactly one edge containing any given pair of nodes 3. You can choose 4 nodes such that no three are contained in a common edge
I'm trying to solve an exercise to prove that for any such object, the number of nodes is $n^2+n+1$ for some n. I pretty much have a proof, which I will post as an answer. However, what I'm really looking for is a cleaner proof or way of thinking about it, as my method seems way too convoluted.