Classify linear equations for which uniformly random two-colorings minimize monochromatic solutions

Determine which nonzero-coefficient linear equations have uniformly random 2-colorings that asymptotically minimize the number of monochromatic solutions.

Background

The paper studies commonness and uncommonness of linear equations under colorings. A linear equation is 2-common over the integers when every 2-coloring asymptotically produces at least as many monochromatic solutions as a uniformly random 2-coloring; otherwise, it is 2-uncommon. The authors state that a general classification of equations satisfying this extremal property is not known.

References

In fact we are still unable to know which linear equations having uniformly random $2$-coloring asymptotically minimize the number of monochromatic solutions.

— On Monochromatic Solutions of Linear Equations Using At Least Three Colors  (2501.17136 - Wijaya, 28 Jan 2025) in Section 1, subsection “Background and Motivation”