Two Multicolor Ramsey Numbers Involving Bipartite Graphs

For graphs G and H, the multicolor Ramsey number \(r_{k,1}(G,H)\) is the minimum N such that any edge-coloring of \(K_N\) by \(k+1\) colors contains a monochromatic G in the first k colors or a monochromatic H in the last color. In this note, we show the asymptotic upper bounds for \(r_{k,1}(K_{m,n};G)\) and \(r_{k,1}(B_{n};K_{t,s})\) if n is large, where G is any graph with at least one edge and \(B_n\) is a book.