In the book "large networks and graph limits", exercise 7.16 is the question in the title.
Let W be a graphon which its all odd cycle homomorphism density is 0.
By proposition 14.21, W is a limit of bipartite graphs. Let their graphon induces bipartitions of [0,1], $(A_n,B_n)$.
By theorem 11.22 and definition of cut distance, we only need to prove: let A,B be upper limits of $A_n^2,B_n^2$.
Then there exists $X,Y\subseteq [0,1]$ such that $X^2\in A,Y^2\in B,m(X\cup Y)=1$. But I can't go any further.
How to solve it?