# combinatorial – Upper limits for the number of \$ 3 \$ -flags in \$ (2k, k ^ 2) \$ – graph \$ G \$

Leave $$G$$ be a simple arbitrary graph in $$2k$$ vertices with $$k ^ 2$$ edges, where $$k geq 2$$. Leave $$F$$ be a $$3$$– Flag, that is, three triangles that share a single border (this graphic has 5 vertices and 7 edges). I want to find an upper limit for the number of $$3$$-flags $$F$$ in $$G$$.

A trivial upper limit is $$(2k) ^ 5$$, but this is too raw and I want an upper limit smaller than this. Any ideas? Thanks in advance!