Ubiquity of power sums in graph profiles

TL;DR

This paper demonstrates that ratios of graph densities often form the power-sum profile in high dimensions, revealing high-dimensional information about graph profiles through simplified extremal problems.

Contribution

It shows that power-sum profiles are prevalent in high-dimensional graph density ratios, linking extremal combinatorics with recent polynomial inequality undecidability results.

Findings

Ratios of graph densities often form the power-sum profile in high dimensions.

Reconstruction of density profiles for 4k-cycles reduces to one-parameter extremal problems.

Power-sum profiles contain high-dimensional information despite not fully determining the density profile.

Abstract

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.