Abstract
Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence G1, G2, …, Gk of pairwise edge-disjoint subgraphs of G such that each subgraph Gi (1≤i≤k) is monochromatic and Gi is isomorphic to a proper subgraph of Gi+1 (1≤i≤k−1). The ascending Ramsey index ARc(G) of G with respect to c is the maximum length of an ascending Ramsey sequence in G with respect to c. The ascending Ramsey index AR(G) of G is the minimum value of ARc(G) among all red-blue colorings c of G. It is shown that there is a connection between this concept and set partitions. The ascending Ramsey index is investigated for some classes of highly symmetric graphs such as complete graphs, matchings, stars, graphs consisting of a matching and a star, and certain double stars.
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