On the 3-Color Ramsey Numbers R(C_4,C_4,W_n)

For given graphs G_1, G_2, … , G_k , k≥ 2 , the k -color Ramsey number, denoted by R(G_1, G_2, … , G_k) , is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of G_i in color i , for some 1 ≤ i ≤ k . Let C_m be a cycle of length m and W_n a wheel of order n+1 . In this paper, we show that R(C_4, C_4, W_ n)≤ n+⌈√(4n+5)⌉ +3 for n=42, 48, 49, 50, 51, 52 or n≥ 56 . Furthermore, we prove that R(C_4, C_4, W_ℓ ^2-ℓ)≤ℓ ^2+ℓ +2 for ℓ≥ 9 , and if ℓ is a prime power, then the equality holds.