Ramsey Theorem
Ramsey Theorem
R(k,)A the smallest integer satisfying: if n=R(k,D),for any 2-coloring of Kn, there exists a red Kk or a blue Ki. 2-coloring of Kn f:(）--od.blwo} Ramsey Theorem R(kl)is finite. Frank P.Ramsey (1903-1930) R(3,3)=6
R(k,l) the smallest integer satisfying: Ramsey Theorem R(k,l) is finite. 2-coloring of Kn f : [n] 2 ⇥ {red, blue} if n≥ R(k,l), for any 2-coloring of Kn, there exists a red Kk or a blue Kl. R(3,3) = 6 Frank P. Ramsey (1903-1930)
If n=Ri(r;k1,k2,...kr), r-colering:(）h ie{1,2,...,r}and Sc[n withS= such that entire is colored by i Theorem(Ramsey 1930) Ri(r;k1,k2,...,kr)is finite
If n ≥ Rt(r; k1, k2, ... , kr), Theorem (Ramsey 1930) Rt(r; k1, k2, ... , kr) is finite. ∀ r-coloring ∃i∈{1,2,...,r} and S⊆[n] with |S|=ki such that entire f : [n] t [r] S t is colored by i
Ifn≥R(r;k≌t(r;k,,) rin:()→月 ョa monochromatic with S[n]and | Theorem(Ramsey 1930) Ri(r;k)is finite
If n ≥ Rt(r; k) Theorem (Ramsey 1930) Rt(r; k) is finite. ∀ r-coloring ∃ a monochromatic with S⊆[n] and |S|=k f : [n] t [r] S t Rt(r; k,..., k r )
Ramsey Theorem Applications
Ramsey Theorem Applications