Sidorenko's Conjecture and Green's Open Problem 12

Stream: graph theory

[Jean-Guillaume Durand]

TLDR: should indices wrap around modulo 5 in Green's Open Problem 12?

I am working on a formalization of Green's Open Problem 12. With @Felix Pernegger and @Moritz Firsching, we are trying to figure out how to interpret the original statement:

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \lbrace i, i+1, i+2 \rbrace$?
Comment: This is very closely related to the Cayley graph case of simplest unknown instance of Sidorenko’s conjecture for graphs, that of the Möbius ladder $K_{5,5} \setminus C_{10}$.

My thoughts so far:

• Interpretation A: consider indices modulo 5. This formulation leads to 15 edges and seems compatible with an expression of Sidorenko's formula on $K_{5,5} \setminus C_{10}$ (see this thread). This is also the interpretation chosen by BuddenBench, and looks like a common setup for Cayley Abelian Graphs (this book p.107). In this case, Green's problem looks like narrowing the Sidorenko conjecture to Cayley graphs on Abelian groups?
• Interpretation B: indices don't wrap around, leading to only 12 edges, not matching Sidorenko's formula with exponent 15 for $\alpha$. Yet, maybe this is what Green means by "closely related"? (i.e. "it is not exactly the Sidorenko conjecture on $K_{5,5} \setminus C_{10}$ for Abelian Cayley graphs")
• Interpretation C: other?

Intuition and the above data points point me towards (A), but I might not be grasping some nuances.
Calling to people familiar with the different flavors of the Sidorenko conjecture: how would you interpret it?

Thank you for the help!

[Bhavik Mehta]

My instinct here was also A

[Felix Pernegger]

Just use A yeah, likely okay (and if nit this can be changed later)

[Kevin Buzzard]

Kevin,

Yes, the indices are to be taken mod 5. I'll update the question.

Best
Ben

P.S. I wasn't aware of the amount of activity on this - just in case it hasn't been noted, I put a new version of the open problems document on my webpage a few days ago, reflecting updates in the last year or so.