Joonkyung Lee (이준경), On graph norms for complex-valued functions

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. there is $\alpha$ such that $\|.\|_{H,\alpha}$) is a norm on the vector space of real-valued (resp. complex-valued) functions. This generalises Gowers 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, by proving that $H$ is complex-norming if and only if it is real-norming. Our proof does not explicitly construct a suitable $2$-edge-colouring $\alpha$ but obtain 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 answers the only question appeared in Hatami’s pioneering work in the area that remained untouched. This is joint work with Alexander Sidorenko.