Extremal Problems and Ramsey Properties of Ball, Box or Orthant containing many points in $R^d$ - And Combinatorics of Permutations(Computational Geometry and Discrete Geometry)
Yoshiyasu Ishigami · Institutional Repositories DataBase (IRDB) · 1994
1 Ball and Box For any points $x=(x_{1}, \cdots, x_{d}),$ $y=(y_{1}, \cdots\cdot, y_{d})\in R^{d}$ , let $Box_{d}(x, y)$ be the smallest d-dimensional standard box in $R^{d}$ which contains the two point $x,$ $y$ , i.e. $Box_{d}(x, y):=$ { $z=(z_{i})_{i}\in R^{d}|x_{i}\leq z_{i}\leq y_{i}$ or $x_{i}\geq z_{i}\geq y_{i}$ for any $1\leq i\leq d$ } $-\{x,y\}$ .And let $Ball_{d}(x, y)$ be the smallest d-dimensinal ball in $R^{d}$ which contains the two points $x,$ $y\in R^{d}$ , i.e. $Ball_{d}(x, y):= \{\frac{1}{2}(x+y)+r|\Vert r\Vert\leq\frac{1}{2}\Vert x-y\Vert\}-\{x, y\}$ , where $\Vert$ .I I means the euclidean norm.For any positive integers $d,$ $n$ , if $F=Box$ or Ball , then we define $\Pi^{F}(n, d)$ the largest number which satisfies the condition $(^{*})$ For any setWhen "For any set $P$ " is replaced by "For any convex set $P$ " in $(^{*})$ , we denote $\Pi^{F}(n, d)$ by $\overline{\Pi}^{F}(n, d)$ .Clearly, $\Pi^{Box}(n, 1)=\overline{\Pi}^{Box}(n, 1)=\Pi^{Ball}(n, 1)=\overline{\Pi}^{Ball}(n, 1)=n$ .Proposition 1 $\Pi^{Box}(n, 2)=[\frac{n-4}{5}t,\overline{\Pi}^{Box}(n, 2)=r\frac{n}{4}t-1$ .