Scholarlink

For integers \(k \ge 3\) , \(c \ge 2\) , and \(s \ge 0\) , let \(M^*_k(c, s)\) be the least integer such that any set of at least \(M^*_k(c, s)\) points in the plane in general position, colored with c colors, contains a monochromatic k-gon (not necessarily convex) with at most s interior points. Denote by \(\lambda _k^*(c)\) the least integer such that \(M^*_k(c, \lambda _k^*(c)) < \infty \) . It follows from results in [3, 5] that \( \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _3^*(c) \le c-3. \) In this paper, we extend this result to \(k \ge 4\) . Specifically, we show that, for \(c\ge 3\) , \( 2 \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _4^*(c) \le 2 c-4. \) Moreover, for \(k \ge 5\) and \(c \ge 2\) , we show that \( (k-2) \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _k^*(c) \le (k-2) c - (k-1). \)

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Almost Empty Monochromatic Polygons in Planar Point Sets

  • 会议论文

  • 发表日期：

    2025年

查看全文
  • Bhaswar B. Bhattacharya,
  • Sandip Das,
  • Sk. Samim Islam,
  • Saumya Sen

摘要

For integers \(k \ge 3\) , \(c \ge 2\) , and \(s \ge 0\) , let \(M^*_k(c, s)\) be the least integer such that any set of at least \(M^*_k(c, s)\) points in the plane in general position, colored with c colors, contains a monochromatic k-gon (not necessarily convex) with at most s interior points. Denote by \(\lambda _k^*(c)\) the least integer such that \(M^*_k(c, \lambda _k^*(c)) < \infty \) . It follows from results in [3, 5] that \( \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _3^*(c) \le c-3. \) In this paper, we extend this result to \(k \ge 4\) . Specifically, we show that, for \(c\ge 3\) , \( 2 \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _4^*(c) \le 2 c-4. \) Moreover, for \(k \ge 5\) and \(c \ge 2\) , we show that \( (k-2) \left\lfloor \frac{c-1}{2} \right\rfloor \le \lambda _k^*(c) \le (k-2) c - (k-1). \)