Ramsey Number r(C_(2m 1), K_n) With Large n

This is an announcement that r(C2m 1, Kn) ＜ c(m) ( ) 1/m has been proved.The Ramsey number r(H, Kn) is the smallest integer N such that every H-free graph onN vertices has independence number at least n. The study of Ramsey number r(Ck, Kn) wasinitiated by Bondy and Erd s2]. They proved that for any fixed n, r(Ck, Kn) = (k - 1)(n - 1) 1if k n2 - 1, and r(Ck, Kn) kn2. For fixed k 3, it is difficult to obtain a satisfied bound ofr(Ck, Kn) for n → ∞ . The bound of Bondy and Erd s w…