Sidorenko's conjecture

Sidorenko's conjecture is a major conjecture in the field of extremal graph theory, posed by Alexander Sidorenko in 1986. Roughly speaking, the conjecture states that for any bipartite graph $H$ and graph $G$ on $n$ vertices with average degree $pn$, there are at least $p^{|E(H)|}n^{|V(H)|}$ labeled copies of $H$ in $G$, up to a small error term. Formally, it provides an intuitive inequality about graph homomorphism densities in graphons. The conjectured inequality can be interpreted as a statement that the density of copies of $H$ in a graph is asymptotically minimized by a random graph, as one would expect a $p^{|E(H)|}$ fraction of possible subgraphs to be a copy of $H$ if each edge exists with probability $p$.