On multicolour noncomplete Ramsey graphs of star graphs

Given graphs G,G1,…,Gk, where k≥2k≥2, the notation G→(G1,G2,…,Gk) denotes that every factorization F1⊕F2⊕⋯⊕Fk of G implies Gi⊆Fi for at least   one ii, 1≤i≤k1≤i≤k. We characterize G for which G→(K(1,n1),K(1,n2),…,K(1,nk)) and derive some consequences from this. In particular, this gives the value of the graph Ramsey number R(K(1,n1),K(1,n2),…,K(1,nk))R(K(1,n1),K(1,n2),…,K(1,nk)).