Sidorenko's conjecture for blow-ups
A celebrated conjecture of Sidorenko and Erd\H{o}s-Simonovits states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion. Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph $H$ with bipartition $A \cup B$ where the number of vertices in $B$ of degree $k$ satisfies a certain divisibility condition for each $k$. As a corollary, we have that for every bipartite graph $H$ with bipartition $A \cup B$, there is a positive integer $p$ such that the blow-up $H_A^p$ formed by taking $p$ vertex-disjoint copies of $H$ and gluing all copies of $A$ along corresponding vertices satisfies the conjecture. Another way of viewing this latter result is that for every bipartite $H$ there is a positive integer $p$ such that an $L^p$-version of Sidorenko's conjecture holds for $H$.