Asymptotic extremal graph theory is non-trivial.

Many fundamental theorems in
extremal graph theory can be expressed as linear inequalities between
homomorphism densities. Lovasz and, in a slightly different formulation,
Razborov asked whether it is true that every such inequality follows
from a finite number of applications of the Cauchy-Schwarz inequality.
In this talk we will show that the answer to this question is negative.
Further, we will show that the problem of determining the validity of a
linear inequality between homomorphism densities is undecidable. Hence
such inequalities are inherently difficult in their full generality.
These results are joint work with Hamed Hatami.

On the other hand,  the Cauchy-Schwarz inequality represents a powerful
tool for obtaining particular approximate and exact results in
asymptotic extremal graph theory. We will mention several results
obtained in this way in joint work with Hatami, Hladky and Kral,
answering questions of Jagger, Stovicek and Thomason, and of Thomasse.