On graph norms for complex-valued functions

TL;DR

This paper unifies the concepts of real- and complex-valued graph norms, proving that a graph is complex-norming if and only if it is real-norming, and applies this to resolve open problems about hypercubes.

Contribution

It establishes a unified framework for real- and complex-valued graph norms, proving their equivalence and providing new insights into which graphs are norming.

Findings

Hypercubes are not norming graphs.

Complex-norming is equivalent to real-norming for graphs.

The proof offers existence and uniqueness of certain edge-colourings.

Abstract

For any given graph $H$, one may define a natural corresponding functional $\|.\|_H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once $H$ is paired with a $2$-edge-colouring $\alpha$ to assign conjugates. We say that $H$ is real-norming (resp. complex-norming) if $\|.\|_H$ (resp. $\|.\|_{H,\alpha}$ for some $\alpha$) is a norm on the vector space of real-valued (resp. complex-valued) functions. These generalise the Gowers octahedral norms, a widely used tool in extremal combinatorics to quantify quasirandomness. We unify these two seemingly different notions of graph norms in real- and complex-valued settings. Namely, we prove that $H$ is complex-norming if and only if it is real-norming and simply call the property norming. Our proof does not explicitly construct a suitable $2$-edge-colouring $\alpha$ but obtains its existence and uniqueness, which may be of independent interest. As an application, we give various example graphs that are not norming. In particular, we show that hypercubes are not norming, which resolves the last outstanding problem posed in Hatami's pioneering work on graph norms.