Signed Ramsey Numbers

Pub Date : 2023-12-28 DOI:10.1007/s00373-023-02736-7

Mohammed A. Mutar, Vaidy Sivaraman, Daniel Slilaty

引用次数: 0

Abstract

Let r(s, t) be the classical 2-color Ramsey number; that is, the smallest integer n such that any edge 2-colored \(K_n\) contains either a monochromatic \(K_s\) of color 1 or \(K_t\) of color 2. Define the signed Ramsey number \(r_\pm (s,t)\) to be the smallest integer n for which any signing of \(K_n\) has a subgraph which switches to \(-K_s\) or \(+K_t\). We prove the following results.

(1)
\(r_\pm (s,t)=r_\pm (t,s)\)
(2)
\(r_\pm (s,t)\ge \left\lfloor \frac{s-1}{2}\right\rfloor (t-1)\)
(3)
\(r_\pm (s,t)\le r(s-1,t-1)+1\)
(4)
\(r_\pm (3,t)=t\)
(5)
\(r_\pm (4,4)=7\)
(6)
\(r_\pm (4,5)=8\)
(7)
\(r_\pm (4,6)=10\)
(8)
\(3\!\left\lfloor \frac{t}{2}\right\rfloor \le r_\pm (4,t+1)\le 3t-1\)

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有符号的拉姆齐数字

让 r(s, t) 是经典的双色拉姆齐数；也就是说，最小的整数 n，使得任何边的双色 \(K_n\) 包含颜色 1 的单色 \(K_s\) 或颜色 2 的单色 \(K_t\)。定义有符号的拉姆齐数（r_\pm (s,t)\）是最小的整数 n，对于这个整数，\(K_n\)的任何符号都有一个切换到\(-K_s\)或\(+K_t\)的子图。我们证明了以下结果。(1)\(r_\pm (s,t)=r_\pm (t,s)\)(2)\(r_\pm (s,t)\ge \left\lfloor \frac{s-1}{2}\right\rfloor (t-1)\)(3)\(r_\pm (s、(4)（r_\pm (3,t)=t）(5)（r_\pm (4,4)=7）(6)（r_\pm (4,5)=8）(7)（r_\pm (4,6)=10）(8)（3\！\left\lfloor \frac{t}{2}\right\rfloor \le r_\pm (4,t+1)\le 3t-1\)

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