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We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of k-edge stars in a graph with density x 0, 1 is asymptotically maximized by a clique and isolated vertices or its complement. Next, among ordered n-vertex graphs with m edges, we determine the maximum and minimum number of copies of a k-edge star whose nonleaf vertex is minimum among all vertices of the star. Finally, for s 2, we define a particular 3-edge-colored complete graph F on 2s vertices with colors blue, green and red, and determine, for each (xb, xg) with xb+xg 1 and xb, xg 0, the maximum density of F in a large graph whose blue, green and red edge sets have densities xb, xg and 1-xb-xg, respectively. These are the first nontrivial examples of colored graphs for which such complete results are proved.
Cairncross et al. (Mon,) studied this question.