# Possible inclusion-exclusion (?) for number of triangles

I am looking for some reasoning in Chatterjee's paper here (page 3) (also picture below) of the bound on the number of triangles in the random graph $$G(n,p)$$ of n verticies and each edge is present with probability p. It somewhat seems like an inclusion exclusion application, but with two conditions (good/bad edges AND verticies) I'm not able to see it. I am also not understanding the significance of $$\epsilon l n p$$ for number of triangles in good edges, since the expectation of good edges is $$(n-2)p^2$$. Simply helping to understand how this inequality exists would be very helpful.

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• It's just a grouping, not a partition or classification, to provide an upper bound. A partition would be for example, triangles with all good edges and triangles with ( not all good edges). The important thing here is to find a triangle of G that does not fall into one of the classes. If you do, you have a basis for complaint. I am not finding a use of inclusion exclusion yet. Gerhard "Not Finding Complaint Basis Either" Paseman, 2019.01.27. – Gerhard Paseman Jan 28 at 5:19