Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from Shkredov's upper bound for the corners problem [Shk06], which shows that $|S| \le O(n^2/(\log \log n)^c)$ for some small $c > 0$, and a construction due to Petrov [Pet23], which shows that $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$. Could it be that for all $\varepsilon > 0$, $|S| \le O(n^{1+\varepsilon})$? We show that if so, this would rule out obtaining $\omega = 2$ using a large family of abelian groups in the group-theoretic framework of Cohn, Kleinberg, Szegedy and Umans [CU03,CKSU05] (which is known to capture the best bounds on $\omega$ to date), for which no barriers are currently known. Furthermore, an upper bound of $O(n^{4/3 - \varepsilon})$ for any fixed $\varepsilon > 0$ would rule out a conjectured approach to obtain $\omega = 2$ of [CKSU05]. Along the way, we encounter several problems that have much stronger constraints and that would already have these implications.

翻译：设 $S \subseteq [n]^2$ 不包含形如 $(x,y), (x,y+\delta), (x+\delta,y')$ 的三点组，其中 $\delta \neq 0$。$S$ 的最大可能大小是多少？显然有 $n \le |S| \le n^2$。基于Shkredov对角落问题[Shk06]的上界估计，可得到这些边界的微弱改进：存在某小常数 $c > 0$ 使得 $|S| \le O(n^2/(\log \log n)^c)$；而Petrov [Pet23] 的构造表明 $|S| \ge \Omega(n \log n/\sqrt{\log \log n})$。是否对所有 $\varepsilon > 0$ 均有 $|S| \le O(n^{1+\varepsilon})$？我们证明：若此结论成立，则基于Cohn、Kleinberg、Szegedy 与 Umans [CU03, CKSU05] 提出的群论框架（该框架已知能捕捉当前 $\omega$ 的最佳上界），将无法通过一大类阿贝尔群实现 $\omega = 2$，且目前尚未发现该框架存在障碍。进一步，若对任意固定 $\varepsilon > 0$ 存在 $O(n^{4/3 - \varepsilon})$ 的上界，则将排除 [CKSU05] 中实现 $\omega = 2$ 的推测性方法。在此过程中，我们遇到了若干约束更强的问题，它们本身已具备这些推论。