Signed Ramsey Numbers

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) $3\!\left\lfloor \frac{t}{2}\right\rfloor \le r_\pm (4,t+1)\le 3t-1$