One color solution

In the mathematical branch of discrete number theory, especially in Ramsey theory , the term monochrome solution describes the property of certain numbers of a colored set of numbers to be colored the same and to satisfy a certain equation . ${\ displaystyle x_ {1} \ ldots x_ {k} \ in [1, n] \ subseteq \ mathbb {N}}$ ${\ displaystyle f (x)}$

definition

Be a - coloring of a set of positive integers and an equation depending on the variables . has a monochrome solution under if and only if there are values for which satisfy and have the same color under . ${\ displaystyle \ chi}$ ${\ displaystyle r}$ ${\ displaystyle f}$ ${\ displaystyle x_ {1} \ ldots x_ {n}}$ ${\ displaystyle \ chi}$ ${\ displaystyle f}$ ${\ displaystyle x_ {1} \ ldots x_ {n}}$ ${\ displaystyle f}$ ${\ displaystyle \ chi}$

properties

The above definition allows the representation , which can be any factors.

{\ displaystyle f: c_ {1} x_ {1} + \ ldots + c_ {n-1} x_ {n-1} = x_ {n}}

{\ displaystyle c_ {i}}

Special cases of have been given a name because of their importance. For example, numbers with x + y = z are called Schur triples . ${\ displaystyle f}$ ${\ displaystyle {x, y, z}}$

For describes a plane in the three-dimensional visual space. ${\ displaystyle n = 3}$ ${\ displaystyle f}$

Examples

The rate of Van der Waerden ensures the existence of van der Waerden numbers , in particular , the number for which the staining with a set of numbers is always an element arithmetic progression is the length of the third We can write these numbers as . We then choose and . The result is the equation with , a plane equation , as a monochrome solution . ${\ displaystyle w (3, r)}$ ${\ displaystyle r}$ ${\ displaystyle w (3, r)}$ ${\ displaystyle \ {a, a + d, a + 2d \}}$ ${\ displaystyle x = a, y = a + 2d}$ ${\ displaystyle z = a + d}$ ${\ displaystyle x + y = 2z}$ ${\ displaystyle x \ not = y}$

Another example and coloring problem of the plane examine the Schur numbers .

Individual evidence

↑ Anusch Taraz: Discrete Mathematics: Fundamentals and Methods . 2012th edition. Birkhäuser, 2012, ISBN 978-3-7643-8898-0 , p. 81 f .

[1] Anusch Taraz: Discrete Mathematics: Fundamentals and Methods . 2012th edition. Birkhäuser, 2012, ISBN 978-3-7643-8898-0 , p. 81 f .