Ubiquity of power sums in graph profiles

Graph density profiles are fundamental objects in extremal combinatorics. Very few profiles are fully known, and all are two-dimensional. We show that even in high dimensions ratios of graph densities and numbers often form the power-sum profile (the limit of the image of the power-sum map) studied recently by Acevedo, Blekherman, Debus and Riener. Our choice of graphs is motivated by recent work by Blekherman, Raymond and Wei on undecidability of polynomial inequalities in graph densities. While the ratios do not determine the complete density profile, they contain high-dimensional information. For instance, to reconstruct the density profile of $4k$-cycles from our results, one needs to solve only one-parameter extremal problems, for any number of $4k$-cycles.