f(8) is at least 4, because ABC+CDE+EFG+GHA can be realized. I don't think it would be too much work to show that 5 is not possible. 1,1,2,2,3,4, unsurprisingly, has hundreds of hits on OEIS, so we need more terms. I have a suspicion that all the graphs of the class you describe (no four vertices with a "lune", AB+AC+BC+BD+CD) are "realizable" as orthogonality graphs in the projective plane. If they are different, then your function f probably won't diverge from the corresponding Ramsey Theory question (largest number of triangles in a luneless graph) for quite a while. We should think about trying to prove the realizability conjecture. I have only the very vaguest inklings about how to try that. On Thu, Jan 30, 2014 at 8:35 PM, Warren D Smith <warren.wds@gmail.com>wrote:
I posted this to math-fun also, but it has not been "approved" yet by RCS:
I can prove f(6)=2 and f(7)=3.
An upper bound on F(n) which shows this is: the maximum number of triangles in an n-vertex graph G, where only G which do not contain <|> as a 4-vertex subgraph, are allowed.
This function of n is also interesting by itself, and I do not know what it is.