TITLE: Looking for history on a theorem of clique intersections
QUESTION [9 upvotes]: 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!

REPLY [11 votes]: 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 .