General Structure Under Forbidden Rainbow Subgraphs

Abstract

The general structure of colored complete graphs containing no copy of a particular rainbow subgraph has been extremely useful in establishing sharp Ramsey-type results for finding monochromatic subgraphs. Several small graphs, like $P_{3}$ for example, immediately trivialize the problem. Indeed, if a colored complete graph contains no rainbow copy of $P_{3}$, then it must be colored entirely with one color. Adding in the third edge to make a triangle already makes the problem much more interesting. When a rainbow subgraph G is forbidden from a coloring, we say the coloring is rainbow G-free . For example, if a rainbow triangle is forbidden, we say the coloring is rainbow triangle-free .