# 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. ${\ displaystyle \ chi}$${\ displaystyle [1, n] \ subseteq \ mathbb {N}}$${\ displaystyle r}$

## 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. ${\ displaystyle [1, r]}$ ${\ displaystyle r}$ ${\ displaystyle \ chi: [1, r] \ rightarrow [1, n] \ subseteq {N}}$${\ displaystyle r}$${\ displaystyle [1, n]}$${\ displaystyle r}$

## properties

• For each color from there is a tuple with . If this is not the case for one , we speak of a coloration.${\ displaystyle i \ in [1, r]}$${\ displaystyle (i, x)}$${\ displaystyle x \ in [1, n]}$${\ displaystyle i}$${\ displaystyle r-1}$
• If there is only one color for each .${\ displaystyle r = 1}$${\ displaystyle n}$
• The number of different stains can easily be obtained by some combinatorial effort.${\ displaystyle r> 1}$
• With the above points it immediately follows that it has to be.${\ displaystyle r \ leq n}$
• 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 ${\ displaystyle r = 3}$${\ displaystyle n = 7}$${\ displaystyle \ chi: \ {1,2,3 \} \ rightarrow \ {1,2,3,4,5,6,7 \}}$

 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. ${\ displaystyle \ chi}$${\ displaystyle 1 \ ldots r}$