Signed Ramsey Numbers

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. (1)

   \(r_\pm (s,t)=r_\pm (t,s)\)

2. (2)

   \(r_\pm (s,t)\ge \left\lfloor \frac{s-1}{2}\right\rfloor (t-1)\)

3. (3)

   \(r_\pm (s,t)\le r(s-1,t-1)+1\)

4. (4)

   \(r_\pm (3,t)=t\)

5. (5)

   \(r_\pm (4,4)=7\)

6. (6)

   \(r_\pm (4,5)=8\)

7. (7)

   \(r_\pm (4,6)=10\)

8. (8)

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