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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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Chromatic number of $C_4$-free graphs

For the second question with respect to maximum degree, a bound of the desired form was first observed by Alon, Krivelevich and Sudakov (see https://doi.org/10.1006/jctb.1999.1910, Cor 2.4). My collea …
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