On graph norms for complex-valued functions

For any given graph (Formula presented.), one may define a natural corresponding functional (Formula presented.) for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once (Formula presented.) is paired with a 2-edge-colouring (Formula presented.) to assign conjugates. We say that (Formula presented.) is real-norming (respectively complex-norming) if (Formula presented.) (respectively (Formula presented.) for some (Formula presented.)) is a norm on the vector space of real-valued (respectively complex-valued) functions. These generalise the Gowers octahedral norms, a widely used tool in extremal combinatorics to quantify quasi randomness. We unify these two seemingly different notions of graph norms in real- and complex-valued settings. Namely, we prove that (Formula presented.) 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 (Formula presented.) 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.