 
Summary: AVOIDING RAINBOW INDUCED SUBGRAPHS IN VERTEXCOLORINGS
MARIA AXENOVICH AND RYAN MARTIN
Abstract. For a fixed graph H on k vertices, and a graph G on at least k vertices, we write G  H
if in any vertexcoloring of G with k colors, there is an induced subgraph isomorphic to H whose
vertices have distinct colors. In other words, if G  H then a totally multicolored induced copy of
H is unavoidable in any vertexcoloring of G with k colors. In this paper, we show that, with a few
notable exceptions, for any graph H on k vertices and for any graph G which is not isomorphic to H,
G H. We explicitly describe all exceptional cases. This determines the induced vertexantiRamsey
number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily
avoidable.
1. Introduction
Let G = (V, E) be a graph. Let c : V (G) [k] be a vertexcoloring of G. We say that G is
monochromatic under c if all vertices have the same color and we say that G is rainbow or totally
multicolored under c if all vertices of G have distinct colors. The existence of a graph forcing an
induced monochromatic subgraph isomorphic to H is well known. The following bounds are due to
Brown and Ršodl:
Theorem 1 (VertexInduced Graph Ramsey Theorem [5]). For all graphs H, and all positive integers
t there exists a graph Rt(H) such that if the vertices of Rt(H) are colored with t colors, then there is
an induced subgraph of Rt(H) isomorphic to H which is monochromatic. Let the order of Rt(H) with
smallest number of vertices be rmono(t, H). Then there are constants C1 = C1(t), C2 = C2(t) such that
