On the boundary of the region defined by homomorphism densities

TL;DR

This paper explores the geometric properties of regions defined by homomorphism densities in graphs, revealing that their boundaries can be highly irregular and not always smooth, unlike the well-understood case of edges and triangles.

Contribution

It extends the understanding of homomorphism density regions by constructing examples where the boundary is nowhere differentiable, showing more complex behavior than previously known.

Findings

Boundary of homomorphism density regions can be non-differentiable.

The well-behaved boundary property does not generalize to arbitrary graph densities.

Constructed examples demonstrate complex boundary structures in higher dimensions.

Abstract

The Kruskal-Katona theorem together with a theorem of Razborov determine the closure of the set of points defined by the homomorphism density of the edge and the triangle in finite graphs. The boundary of this region is a countable union of algebraic curves, and in particular, it is almost everywhere differentiable. One can more generally consider the region defined by the homomorphism densities of a list of given graphs, and ask whether the boundary is as well-behaved as in the case of the triangle and the edge. Towards answering this question in the negative, we construct examples which show that the restrictions of the boundary to certain hyperplanes can have nowhere differentiable parts.