 
Summary: ON COLORINGS AVOIDING A RAINBOW CYCLE AND A FIXED
MONOCHROMATIC SUBGRAPH
MARIA AXENOVICH AND JIHYEOK CHOI
Abstract. Let H and G be two graphs on fixed number of vertices. An edge coloring of a complete graph is
called (H, G)good if there is no monochromatic copy of G and no rainbow (totally multicolored) copy of H in
this coloring. As shown by Jamison and West [14], an (H, G)good coloring of an arbitrarily large complete
graph exists unless either G is a star or H is a forest. The largest number of colors in an (H, G)good
coloring of Kn is denoted maxR(n, G, H). For graphs H which can not be vertexpartitioned into at most
two induced forests, maxR(n, G, H) has been determined asymptotically, [3]. Determining maxR(n; G, H)
is challenging for other graphs H, in particular for bipartite graphs or even for cycles. This manuscript
treats the case when H is a cycle. The value of maxR(n, G, Ck) is determined for all graphs G whose edges
do not induce a star.
1. Introduction and main results
For two graphs G and H, an edge coloring of a complete graph is called (H, G)good if there is no
monochromatic copy of G and no rainbow (totally multicolored) copy of H in this coloring. The mixed
antiRamsey numbers, maxR(n; G, H), minR(n; G, H) are the maximum, minimum number of colors in an
(H, G)good coloring of Kn, respectively. The number maxR(n; G, H) is closely related to the classical
antiRamsey number AR(n, H), the largest number of colors in an edgecoloring of Kn with no rainbow copy
of H introduced by Erdos, Simonovits and S´os [9]. The number minR(n; G, H) is closely related to the
classical multicolor Ramsey number Rk(G), the largest n such that there is a coloring of edges of Kn with k
