The s-chromatic Ramsey number for stars

In 1977, Chung, Chung and Liu generalized the definition of the Ramsey number. They introduced the s-chromatic Ramsey number as follows. Let 1≤ s< t be integers and let A₁, A₂, , Ac be subsets with size s of $[t]$, where c= t s. For given graphs G₁, G₂, , Gc, the { s-chromatic Ramsey number} rs, t(G₁, G₂, , Gc), is the minimum positive integer N such that every t-coloring of E(KN) yields a copy of Gᵢ whose edges are colored by colors in the color set Aᵢ for some i∈ [c]. The { star-critical s-chromatic Ramsey number} r*s, t(G₁, G₂, , Gc), is the minimum integer such that every t-coloring of the edges in KN- E(K1, N- 1-) yields a copy of Gᵢ whose edges are colored by colors in the color set Aᵢ for some i∈ [c], where N= rs, t(G₁, G₂, , Gc). If G₁= G₂= = Gc= G, then we simplify them to rs, t(G) (also called the { weakened Ramsey number}) and rs, t*(G), respectively. In this paper, we determine all the values of rs, t(K1, m) and r*s, t(K1, m), and part of the value of rs, t(K_1, m₁, K_1, m₂, , K_1, mc).