Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets

Fix one colour class in the checkerboard colouring of an $n\times n$ integer grid, and let $M_4(n,\varepsilon)$ be the largest subset having at most two points in every row, column, and diagonal of slopes $\pm1$. We prove the near-saturation bound $M_4(n,\varepsilon)\leq2n-4$ for $n\geq6$. The proof