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!



Looking for history on a theorem of clique intersections
Hajnal's Clique Collection Lemma implies your theorem. See Corollary 2.10 from this paper <http://arxiv.org/pdf/1101.4564v5.pdf> or Hajnal's original paper <http://cms.math.ca/cjm/v17/cjm1965v17.0720-0724.pdf> .


