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Study of graphs satisfying a property that are maximal or minimal with respect to some parameter. A classic example is Turán's Theorem, which exactly characterizes the densest graphs on $n$ vertices without a $K_t$ subgraph.

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Random graphs require O(n log(n)) edges until they are almost certainly fully connected - wh...

according to the double exponential theorem of Erdos and Renyi, I would say that you need to add $(n \ln n)/2 + n Z_n /2$ edges in which $Z_n$ is a random variable that converges in distribution to th …
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