Ifn≥R(r;k1,k2,…,kr), for any r-partition of ()=CU..UCr, there exists an S C [n]such that Sk and()Ci for some i∈{1,2，.，r}. mixing color: Rr;k1,…,k-2,kl,)≤R(r-1;k1,…,k-2,R(2；kl,k) R(k,)≤R-(R(k-1,),R(k,l-1)+1 n=R-(R(k-1,),R(k,l-1)+1 goal:Vf (){red,blue} (),X1=kor(),1Y1=1If n ≥ Rt(r; k1, k2, ... , kr), Rt(r; k1, ... , kr-2, kr-1, kr) ≤ Rt(r-1; k1, ... , kr-2, Rt(2; kr-1, kr)) mixing color: Rt(k,l) ≤ Rt-1(Rt(k-1,l), Rt(k,l-1)) + 1 n = Rt-1(Rt(k-1,l), Rt(k,l-1)) + 1 goal: ⇥f : ￾[n] t ⇥ ￾ {red, blue} ￾ ￾X t ⇥ , |X| = k or ￾Y t ⇥ , |Y | = l for any r-partition of ￾[n] t ￾ = C1[···[Cr, there exists an S ✓ [n] such that |S| = ki and ￾S t ￾ ✓ Ci for some i 2 {1, 2,...,r}