On the multi-colored Ramsey numbers of cycles

For a graph L and an integer k≥2, Rk(L) denotes the smallest integer N for which for any edge-coloring of the complete graph KN by k colors there exists a color i for which the corresponding color class contains L as a subgraph. Bondy and Erdo″s conjectured that, for an odd cycle Cn on n vertices, Rk(Cn)=2 k-1(n-1)+1 for n>3. They proved the case when k = 2 and also provided an upper bound Rk(Cn)≤(k+ 2)!n. Recently, this conjecture has been verified for k = 3 if n is large. In this note, we prove that for every integer k≥4, Rk(Cn≤ k2kn+o «n» as n → ∞ When n is even, Sun Yongqi, Yang Yuansheng, Xu Feng, and Li Bingxi gave a construction, showing that Rk(C n≥(k-1)n-2k+ 4. Here we prove that if n is even, then R k(Cn≤kn+o(n) as n→∞.