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 Kn contains either a monochromatic Ks of color 1 or Kt of color 2. Define the signed Ramsey number r±(s, t) to be the smallest integer n for which any signing of Kn has a subgraph which switches to - Ks or + Kt . We prove the following results. (1)r±(s, t) = r±(t, s) (2)r±(s,t)≥⌊s-12⌋(t-1)(3)r±(s, t) ≤ r(s- 1 , t- 1) + 1 (4)r±(3 , t) = t(5)r±(4 , 4) = 7 (6)r±(4 , 5) = 8 (7)r±(4 , 6) = 10 (8)3⌊t2⌋≤r±(4,t+1)≤3t-1.