Determine whether 2-commonness implies r-commonness

Determine whether every linear equation that is 2-common over the integers is also r-common over the integers for every integer r at least 3.

Background

The paper proves that every 2-uncommon linear equation is r-uncommon for every r at least 3. It contrasts this result with the unresolved converse-direction behavior for equations known to be 2-common. In particular, the equation with equal sums of m variables, x1+⋯+xm=xm+1+⋯+x2mx_1+\dots+x_m=x_{m+1}+\dots+x_{2m}, is known to be 2-common over the integers, but its 3-commonness is not established even when m equals 2.

References

We showed that $2$-uncommonness implies $r$-uncommonness for any $r\geq 3$, but we still do not know whether if an equation is $2$-common then it is also $r$-common. Even in the simplest case where $m=2$ in eq:even we are not able yet to determine whether it is $3$-common or not.

— On Monochromatic Solutions of Linear Equations Using At Least Three Colors  (2501.17136 - Wijaya, 28 Jan 2025) in Section 4, “Open Problem”