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

Abstract

Fix one colour class in the checkerboard colouring of an $n\times n$ integer grid, and let $M_4(n,\epsilon )$ be the largest subset having at most two points in every row, column, and diagonal of slopes $\pm 1$. We prove the near-saturation bound $M_4(n,\epsilon )\le 2n-4$ for $n\ge 6$. The proof uses first and second moments of the four line families: a hypothetical set of size $2n-3$ produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy--Schwarz. A finite argument handles $n=6$. The same identity extends to arbitrary deficit multisets. It gives $M_4(n,\epsilon )\le 2n-d$ whenever $d\ge 4$ is an integer and $n\ge 3d-4$, and an entirely discrete asymptotic estimate \[ M_4(n,\varepsilon)\leq(\sqrt{21}-3)n+8. \] We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, \[ L_{\mathrm{mono}}(n,\varepsilon)\leqαn+O(1), \qquad α\approx1.5768233968738, \] and hence the same upper bound for $M_4$ and for checkerboard no-three-in-line sets.

18 pages