Multicolored Bipartite Ramsey Numbers of Large Cycles

For an integer r ≥ 2 and bipartite graphs H_i, where 1≤ i ≤ r the bipartite Ramsey number br(H₁, H₂, …, H_r) is the minimum integer N such that any r-edge coloring of the complete bipartite graph K_N,N contains a monochromatic subgraph isomorphic to H_i in color i for some 1 ≤ i ≤ r. We show that if \(r \ge 3,{\alpha _1},{\alpha _2} > 0,{\alpha _{j + 2}} \ge [(j + 2)! - 1]\sum\limits_{i = 1}^{j + 1} {{\alpha _i}} \) for j = 1, 2, …, r −2, then \(br({C_{2\left\lfloor {{\alpha _1}\,n} \right\rfloor }},{C_{2\left\lfloor {{\alpha _2}\,n} \right\rfloor }}, \cdots ,{C_{2\left\lfloor {{\alpha _r}\,n} \right\rfloor }}) = (\sum\limits_{j = 1}^r {{\alpha _j} + o(1))n} \).