# theoretical measurement – Why is a graphon bipartite if and only if its odd cycle density is 0?

In the book "Great Networks and Graphical Boundaries", Exercise 7.16 is the question in the title.

Let W be a graphon whose odd-cycle homomorphism density is 0.

By proposition 14.21, W is a limit of bipartite graphs. Let their graphon induce bipartitions of [0,1], $$(A_n, B_n)$$.

According to Theorem 11.22 and the definition of the cutting distance, it suffices to prove that: A and B are the upper limits of $$A_n ^ 2, B_n ^ 2$$.

So there is $$X, Y subseteq [0,1]$$ such as $$X ^ 2 in A, Y ^ 2 in B, m (X cup Y) = 1$$. But I can not go further.

How to solve it?