发表机构
University of Toronto; School of Mathematical Sciences, Queen Mary University of London; University of the Fraser Valley（多伦多大学; 伦敦玛丽女王大学数学科学学院; 弗雷泽河谷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对棋盘无三点共线集合问题，通过矩方法证明近饱和上界，并利用连续对偶证书导出渐近常数约1.5768的线性界。
AI 中文摘要
固定$n\times n$整数网格棋盘着色中的一个颜色类，令$M_4(n,\varepsilon)$为在每一行、每一列以及斜率为$\pm1$的每条对角线上至多有两个点的最大子集。我们证明了近饱和界$M_4(n,\varepsilon)\leq2n-4$（对$n\geq6$成立）。证明使用了四条线族的矩：一个大小为$2n-3$的假设集合产生的行、列和对角线亏缺，其精确矩恒等式与柯西-施瓦茨不等式矛盾。有限论证处理$n=6$的情况。同一恒等式可推广到任意亏缺多重集。它给出$M_4(n,\varepsilon)\leq2n-d$，其中$d\geq4$为整数且$n\geq3d-4$，以及一个完全离散的渐近估计\\[ M_4(n,\varepsilon)\leq(\sqrt{21}-3)n+8. \\]我们还证明了相关四方向分数打包问题的一般连续到离散定理。将其应用于先前工作中构造的精确连续对偶证书，对两种颜色均得到\\[ L_{\mathrm{mono}}(n,\varepsilon)\leq\alpha n+O(1), \qquad \alpha\approx1.5768233968738, \\]从而得到$M_4$和棋盘无三点共线集合的相同上界。
英文摘要
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 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 $M_4(n,\varepsilon)\leq2n-d$ whenever $d\geq4$ is an integer and $n\geq3d-4$, 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.
Comments18 pages
