Published online by Cambridge University Press: 12 March 2014
Let κ be an uncountable cardinal and the edges of a complete graph with κ vertices be colored with ℵ0 colors. For the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if , then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on vertices are colored with ℵ0 colors. In particular, we are concerned with the case ℵ1 ≤ κ ≤ ℵω.
The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models . When the graph being colored has size ℵ1, the identities that must occur have been classified by Shelah . If the graph has size greater than or equal to ℵω the identities have also been classified in . The number of colors is fixed at ℵ0 as it is the natural place to start and the results here can be generalized to situations where more colors are used.
There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on vertices. First consider the branches in the complete binary tree of height ω to be vertices of a complete graph.