if n=R(k,D,for any 2-coloring of Kn, there exists a red Kk or a blue Ki. R(k,)≤R(k,l-1)+R(k-1,) take n R(k,1-1)+R(k-1,1) arbitrary vertex v ISI+I71+1=n R(k,1-1)+R(k-1,1) N≥R) Kkin S Kil in S or Ki T≥Rk1,0rKmT K Kiin T
S T if n≥ R(k,l), for any 2-coloring of Kn, there exists a red Kk or a blue Kl. R(k,l) ≤ R(k,l-1) + R(k-1,l) v take n = R(k,l-1) + R(k-1,l) arbitrary vertex v |S| + |T| + 1 = n = R(k,l-1) + R(k-1,l) |S| ≥ R(k,l-1) |T| ≥ R(k-1,l) or or Kk in S Kl-1 in S or Kk-1 in T Kl in T +v Kl +v Kk
if n=R(k,l),for any 2-coloring of Kn, there exists a red Ki or a blue Ki. R(k2)=k;R(2,)=I; R(k,)≤R(k,l-1)+R(k-1,) Ramsey Theorem R(k,is finite. By induction: k+1-2 R(k,)≤（k-1
if n≥ R(k,l), for any 2-coloring of Kn, there exists a red Kk or a blue Kl. R(k,2) = k ; R(2,l) = l ; R(k,l) ≤ R(k,l-1) + R(k-1,l) Ramsey Theorem R(k,l) is finite. R(k,l) ⇥ k + l 2 k 1 By induction: ⇥