I'm very new to graph theory, so apologies if the language isn't quite there, or it's a repeat question. I've looked around, but not knowing the language it's hard to navigate.
I want to choose n-1 not necessarily distinct, subgraphs of the complete graph with n vertices, such that the subgraphs are all complete graphs of order 1≤k≤n. Then I want to prove that you can colour one or more of the n vertices red or blue or leave it blank such that each subgraph is either uncoloured, or contains an edge with a red and blue vertex on either end. I think this might have something to do with Ramsey theory, but I don't really understand that yet.
I adapted this from a question in set theoretic notation, that being let A1,...,An-1 be non empty subsets of {1,...,n} and prove that there is a red/blue/blank colouring of {1...n} such that Ai is either uncolored or contains red and blue for all i. I think I adapted it correctly into graph theory, but I'm not sure if that's the case.
Any help is appreciated