Bipartite Ramsey numbers of large cycles

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\ldots,H_r)$ 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 $i$, $1\leq i\leq r$. We show that for $\alpha_1,\alpha_2>0$, $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor})=(\alpha_1+\alpha_2+o(1))n$. We also show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum^{j+1}_{i=1} \alpha_i$ for $j=1,2,\ldots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\ldots,C_{2\lfloor \alpha_r n\rfloor})=(\sum^r_{j=1} \alpha_j+o(1))n.$ For $\xi>0$ and sufficiently large $n$, let $G$ be a bipartite graph with bipartition $\{V_1,V_2\}$, $|V_1|=|V_2|=N$, where $N=(2+8\xi)n$. We prove that if $\delta(G)>(\frac{7}{8}+9\xi)N$, then any $2$-edge coloring of $G$ contains a monochromatic copy of $C_{2n}$.Comment: 19 page