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 ${\displaystyle H}$ and graph ${\displaystyle G}$ on ${\displaystyle n}$ vertices with average degree ${\displaystyle pn}$, there are at least ${\displaystyle p^{|E(H)|}n^{|V(H)|}}$ labeled copies of ${\displaystyle H}$ in ${\displaystyle G}$, 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 ${\displaystyle H}$ in a graph is asymptotically minimized by a random graph, as one would expect a ${\displaystyle p^{|E(H)|}}$ fraction of possible subgraphs to be a copy of ${\displaystyle H}$ if each edge exists with probability ${\displaystyle p}$.