Tilted Corners in Integer Grids

The number of $k$-dimensional corner-free subsets of grids

A subset $A$ of the $k$-dimensional grid $\{1,2, \cdots, N\}^k$ is called $k$-dimensional corner-free if it does not contain a set of points of the form $\{ a \} \cup \{ a + de_i : 1 \leq i \leq k \}$ for some $a \in \{1,2, \cdots, N\}^k$ and $d > 0$, where $e_1,e_2, \cdots, e_k$ is the standard basis of $\mathbb{R}^k$. We define the maximum size of a $k$-dimensional corner-free subset of $\{1,2, \cdots, N\}^k$ by $c_k(N)$. In this paper, we show that the number of $k$-dimensional corner-free subsets of the $k$-dimensional grid $\{1,2, \cdots, N\}^k$ is at most $2^{O(c_k(N))}$ for infinitely m…

Escaping from the corner of a grid by edge disjoint paths

Let $Q$ be a finite subgraph of the integer grid $G$ in the plane, and let $T$ be a set of pairs of distinct vertices in $G$, called `terminal pairs'. Escaping a subset $X\subset T\cap Q$ from $Q$ means finding edge disjoint paths from the terminals in $X$ into distinct vertices of a set $L$ in the boundary of $Q$. Here we prove several lemmas for the cases where $Q$ is a $3\times 3$ grid, $L$ is the union of a vertical and horizontal boundary line of $Q$, furthermore, $T$ is a set of four terminal pairs in $G$ such that $|T\cap Q|\geq 5$. These lemmas are applied in [4] and complete the proof…

存在障碍物的网格上的最大通行点

Points of maximal traffic on a grid with obstruction

该研究探讨存在障碍物B的n×n网格上，避开B的格路的最大通行点位置，发现n≥9时最大值集中在近端点的10个点，而B在反对角线x+y=n上时会出现异常迁移，推测n≥496时该异常消失。