There has recently been a lot of interplay between extremal graph theory and the theory of graph limits. After a brief survey of the theory of dense graph limits, we focus on exploring a link between Sidorenko's Conjecture, one of the most prominent open problems in extremal graph theory, and weakly norming graphs, one of the concepts studied in the theory of graph limits. Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We are interested in a stronger property, which we call the step Sidorenko property. We relate this property to weakly norming graphs and use our results to construct a bipartite edge-transitive graph that is not weakly norming - this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150].
The talk is based on joint work with Taisa Martins, Peter Pal Pach and Marcin Wrochna.