Journal of Combinatorics

Volume 10 (2019)

Number 2

On the boundary of the region defined by homomorphism densities

Pages: 203 – 219



Hamed Hatami (School of Computer Science, McGill University, Montreal, Quebec, Canada)

Sergey Norin (Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)


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.

The authors were supported by an NSERC grant.

Received 27 April 2017

Published 25 January 2019