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Reasoning behind the safety oracle that uses Turan's Theorem. Can be found here: https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem
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| Turán's Theorem: Let G be any graph with n vertices, such that G is K_{r+1}-free - that is, it has no clique of size r+1 or greater. Then the number of edges in G is at most: | |
| (r-1) / r * n^2 / 2 | |
| Let us say that we have e edges in our graph, and solve for r | |
| e = (r-1) / r * n^2 / 2 | |
| er = (r-1) * n^2 / 2 | |
| er = rn^2 / 2- n^2 / 2 | |
| er - rn^2 / 2= - n^2 / 2 | |
| r (e - n^2/2) = - n^2 / 2 | |
| r (n^2/2 - e) = n^2 / 2 | |
| r = n^2 / (2(n^2/2 - e)) | |
| r = n^2 / (n^2 - 2e) | |
| Thus, we can say that in a graph w/ n nodes and e edges, there is at least a clique of size ceiling(n^2 / (n^2 - 2e)). |
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Throw in an extra set of parenthesis for the eyes
e = [(r-1) / r] * [n^2 / 2]