Ramsey chains in galaxies

Every red-blue coloring of the edges of a graph G results in a sequence G1, G2, . .., G of pairwise edge-disjoint monochromatic subgraphs Gi (1 i ) of size i such that Gi is isomorphic to a subgraph of Gi+1 for 1 i -1.Such a sequence is called a Ramsey chain in G and ARc(G) is the maximum length of a Ramsey chain in G with respect to a red-blue coloring c.The Ramsey index AR(G) of G is the minimum value of ARc(G) among all red-blue colorings c of G.A galaxy is a graph each of whose components is a star.For a positive integer s, an s-galaxy consists of s stars.Thus, a 1-galaxy of size m is the star Sm and an m-galaxy of size m is the matching mK2.A 2-galaxy Sx + Sm-x is a galaxy consisting of the two stars Sx and Sm-x for positive integers x and m 2. The Ramsey indexes of 1-galaxies and m-galaxies are known as well as all 2-galaxies Sx + Sm-x for integers x = 1, 2 and m 3. Here, the exact values of AR(Sx + Sm-x) are determined for integers x = 3, 4 and m 6.Additional results and conjectures on Ramsey chains in 2-galaxies are also presented.