Multicolor Ramsey theory for a fan versus complete graphs

Let mK2 be the disjoint union of m copies of K2 and define the fan Fm := K1 + mK2.The multicolor Ramsey number, r(G1, G2, . . ., Gt) is the least natural number p such that any t-coloring of Kp contains a monochromatic Gi in some color i.The star-critical Ramsey number is denoted by r * (G1, G2. . . ., Gt).We consider the case of fans versus complete graphs.We show that r(Fm, K3, K3) = 10m + 1 and r * (Fm, K3, K3) ≥ 8m + 2, for all m ≥ 6.We also examine the corresponding Gallai-Ramsey numbers, proving that gr(F2, K3, K3) = 21 and gr * (F2, K3, K3) = 18.