I have a short paper I'm working on where I prove:
Theorem: Every graph on (2t-1) vertices with no (t+1)-clique has a vertex that is contained in every t-clique.
By "t-clique", I mean a complete subgraph with t vertices. It's actually a lemma for the main result in the paper, but it's where most of the work is done. It feels like something that has been done before, but I haven't come up with anything in my searches. I think it might be related to clique graphs or clique-helly graphs, but I haven't found a direct link.
Does anyone know of a paper with this result? Or another result that would imply this one? Or at least a related paper that might be worth looking into? Thanks!