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!