5. $B+C~$Conjecture

In [MR0472752], Erdős conjectured that if $A\subseteq\mathbb{N}$ has positive lower asymptotic density (in $\mathbb{N}$), then there are infinite subsets $B,C\subseteq\mathbb{N}$ such that $B+C\subseteq A$.
An early partial result is:Theorem.
(Nathanson [MR0563552]) If $A\subseteq\mathbb{N}$ has positive upper density then, for any $n>0$, there are $B,C\subseteq\mathbb{N}$ such that $B$ is infinite, $C$ has size at least $n$, and $B+C\subseteq A$.
Further progress was made in the setting of countable amenable groups.Theorem.
(Di Nasso, Goldring, Jin, Leth, Lupini, Mahlburg [MR3361013]) Let $G$ be a countable amenable group. If $A\subseteq G$ has positive Banach density then there are infinite subsets $B,C\subseteq G$ and $h_1,h_2\in G$ such that $B\cdot C\subseteq h_1A\cup h_2A$.
 If $A\subseteq G$ has Banach density greater than $\frac{1}{2}$ then there are infinite subsets $B,C\subseteq G$ such that $B\cdot C\subseteq A$.
Problem 5.1.
[Isaac Goldbring] Suppose $G$ is a countable amenable group and $A\subseteq G$ has positive Banach density. Are there infinite $B,C\subseteq G$ such that $B\cdot C\subseteq A$? 
Let $\mathcal{M}$ be a firstorder structure with universe $M$. Given, $m,n>0$, we say a definable subset $R\subseteq M^m\times M^n$ has the independence property if for all $k>0$, there are $(a_i)_{i\in\{1,\ldots,k\}}\subseteq M^m$ and $(b_I)_{I\subseteq\{1,\ldots,k\}}\subseteq M^m$ such that $(a_i,b_I)\in R$ if and only if $i\in I$.
For simplicity, we say “$G$ is an expansion of a group" two mean that $G$ is a firstorder structure containing a binary operation satisfying the group axioms. By imposing various notions of “model theoretic tameness" on $G$, we obtain results on the $B+C$ conjecture for definable sets.Theorem.
(Andrews, Conant, Goldbring [arXiv:1701.07791]) Let $G$ be an expansion of a countable amenable group, and fix a definable set $A\subseteq G$ with positive Banach density. If $G$ is stable then there are infinite $B,C\subseteq G$ such that $B\cdot C\subseteq A$.
 If $G$ is distal with elimination of $\exists^\infty$ then there are infinite definable $B,C\subseteq G$ such that $B\cdot C\subseteq A$.
A structure $\mathcal{M}$ is NIP if, for all $m,n>0$, no definable subset of $M^m\times M^n$ has the independence property.Problem 5.2.
[Isaac Goldbring] Suppose $G$ is an NIP expansion of a countable amenable group. Let $A\subseteq G$ be definable, with positive upper Banach density. Are there infinite $B,C\subseteq G$ such that $B\cdot C\subseteq A$?
Remark. [org.aimpl.user:gconant@nd.edu] During the workshop, our group made the following progress: Suppose $G$ is a (not necessarily countable) abelian group and $A\subseteq G$ is definable in an NIP expansion of $G$. If $A$ has positive Banach density then there are infinite $B,C\subseteq A$ such that $B+C\subseteq A$. In fact, it is enough to just assume that the formula $x+y\in A$ is NIP, which is equivalent to saying that the family of translates $\{g+A:g\in G\}$ has finite VCdimension.
Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.