Three Color Ramsey Numbers R(C_m1,C_m2,C_m3)

Ramsey theory constitutes the main research area of graph theory, and 3-coloring Ramsey theory is an important branch of Ramsey theory, the determination of 3-coloring Ramsey number is also an important research direction of Ramsey theory, and it is meanwhile a No-Polynomial Problem. Ramsey theory has great theoretical significance in the evolution of mathematics, however, we only obtained a few of 3-coloring Ramsey numbers so far.Let G_i be the subgraph of G whose edges are in the i -th color in an r -coloring of theedges of G . If there exists a r -coloring of the edges of G such that forbidden subgraph H_i (?) G_i, for all 1≤i≤r, then G is said to be r-colorable to(H₁,H₂,...,H_r). The multicolorRamsey number R(H₁H₂,...,H_r) is the smallest integer n such that K_n is not r-colorable to(H₁,H₂,...,H_r). The Ramsey number R(C_m1,C_m2,C_m3) is studied in this paper, where r = 3and H(?) C.Erd6s proved that R{C_m, C₃, C₃) = 5m - 4 and R(C_m, C₄, C₄) = m + 2 when m is sufficiently large.The paper proves that R(C_m,C₃,C₃) = 5m-4 when m≥5; When m₁, is not sufficientlylarge and 7≥m₁>m₂l≥m₃, this paper uses the concept of critical graph to obtain all the values of Ramsey number R(C_m1,C_m2,C_m3), meanwhile gives a conjecture. After giving a new definition of critical graph, this paper also obtains all the values of Ramsey number R(C_m,C₄, C₄), what is R(C_m,C₄,C₄) = m + 2 when 11≤m≤19, and also prove that R(C_m,C₄,C₄) =m + 2 when m>19.