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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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maximal sets of vertices that avoids a clique

Given the fixed $k$, you could look at all $\binom{n}{k}$ subsets of vertices to see if they form a clique of size $k$ (or do whatever you like to enumerate all cliques of size $k$ -- there might be a …
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