ORDERED AND COLORED SUBGRAPH DENSITY PROBLEMS

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 [Studia Sci. Math. Hungar., 55 (2018), pp. 238--259] and prove that the number of k-edge stars in a graph with density x \in [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 \geq 2, we define a particular three-edge-colored complete graph F on 2s vertices with colors blue, green, and red and determine, for each (xb,xg) with xb+xg \leq 1 and xb,xg \geq 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.