Abstract
For graphs G and H, denote by rk+1(G;H) the minimum N such that any edge-coloring of KN by k+1 colors contains either a monochromatic G in the first k colors or a monochromatic H in the last color. As usual, we write r2(G;H) as r(G,H). We show that if integers s≥t≥m≥1, then rk+1(Kt,s;Km,n)≤n+(1+o(1))(s−t+1)1/tkmn1−1/t as n→∞. The upper bound is shown to be sharp up to the asymptotical sub-linear term for rk+1(K2,s;K1,n), rk+1(K3,3;K1,n), r(K2,s,K2,n) and r(K3,3,Km,n) for m≤3.
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