Let $A,B,C \subseteq \mathbb{Z}_n$. Suppose that for any $a' \in A, b' \in B, c' \in C$,
\begin{align*} |(A+b') \cap (B+a') \cap -C| &\le 1,\\ |(A+c') \cap -B \cap (C+a')| &\le 1,\\ |-\hspace{-0.15cm}A \cap (B+c') \cap (C+b')| &\le 1. \end{align*} In other words, for any fixed $a', b'$, the system of equations $a'+b+c=a+b'+c=0$ has at most one solution, and the symmetric conditions hold for fixed $a',c'$ and $b',c'$.
Question 1: Subject to these constraints, how many solutions to $a+b+c=0$ can there be, with $a \in A, b \in B, c \in C$?
The answer is at least $n$, obtained for instance by $A=\{0\},B=C=\mathbb{Z}_n$. By an averaging argument and Cauchy–Schwartz, the answer is at most $n^{3/2}$. I think that with the arithmetic regularity lemma, I can improve this to $o(n^{3/2})$. But I am wondering if the much better upper bound of $O(n)$, or even $n$, actually holds. I've checked with my computer that for $n \le 13$, the lower bound of $n$ is best possible.
Question 1 is essentially equivalent to the following. Let $T_n$ be a triangular array of $n(n+1)/2$ points in the plane. Let $L$ be a collection of lines parallel to the sides of the array such that for every four points in $T_n$ forming an equilateral trapezoid with sides parallel to the sides of the array, at least one of its vertices is contained in a line in $L$. Equivalently, after deleting these lines, we are left with a set free of such trapezoids. Here my definition of "equilateral trapezoid" includes the degenerate case of a triangle; see the picture on the left below for an example of some equilateral trapezoids in $T_8$.
Question 2: What is the maximum over all such $L$ of the number of points not contained in any line, that is, $|T_n - \cup_{\ell \in L} \ell|$?
You can trivially achieve $n$ by taking all but one line in one direction, as in the picture on the above right. Is $n$ optimal? If not, is the answer $O(n)$?
Question 3: Is the answer $O(n)$ even for the version of the problem where you just have to cover one vertex of every triangle with sides parallel to the sides of the array?
The best upper bound I know for the third question is $o(n^2)$, by the corners theorem of Ajtai and Szemeredi. However, I don't see how to use current constructions of corner-free sets, as only being allowed to delete lines in these 3 directions seems to be very restrictive.
