# Coloring (number theory)

Under a **coloring** is understood in the discrete number theory coloring a set of numbers with colors . The coloring of sets of numbers is mainly used in Ramsey theory , which examines under certain conditions to what extent regularities can be found in colored subsets.

## definition

Be different colors. The mapping defines a so-called coloration on a subset of the positive integers , through which each element of the subset is assigned one of the colors.

## properties

- For each color from there is a tuple with . If this is not the case for one , we speak of a coloration.
- If there is only one color for each .
- The number of different stains can easily be obtained by some combinatorial effort.
- With the above points it immediately follows that it has to be.
- The coloring of the number is always arbitrary. For this reason, the concept of coloring is used in Ramsey theory , which tries to find out conditions for certain regularities for colored subsets.

## example

We choose and . There are these numbers several colorations possible for would

1 | 2 | 3 | 4th | 5 | 6th | 7th |

R. | B. | G | B. | R. | R. | G |

While the definition of colors is used, concrete examples of these i. d. R. Colors such as *red, green, blue* are used.