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Let $C_m$ denote the cycle on $m\\ge4$ vertices and let $\\Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. Unlike Ramsey number of odd cycles, little is known about the general behavior of $R_k(C_{2n})$ except that $R_k(C_{2n})\\ge (n-1)k+n+k-1$ for all $k\\ge2$ and $n\\ge2$. 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