Schur's theorem

The set of Schur delivered in discrete mathematics statements, how large a set of numbers must be in order for any staining this always exists a monochromatic solution. This set was originally a lemma in a paper by Isay Schur in 1916. This virgin was not out to investigate the coloring of points in the plane, but Fermat's last theorem ( who races through a proof in 1995 was at the rate ). Although found there for twelve years Ramsey, he is regarded as the first set of Ramsey theory.

Wording of the sentence

Background

In block coloring problems the plane are considered. Be a simple level and the set of all points of the plane with positive integer coordinates. For example, and, this time, and need not be forcibly different. Now a finite set colors will be selected and each positive integer assigned a color.

After all the points are then inked exactly with the appropriate color if the color is identical to and on the number line. All such unrecognized items are marked with a. There remains the question whether the existence of a colored point is backed up, or there is a possibility to mark each point of the plane with one. In other words, whether or not a color exists on, so that no dot is colored. This question is answered the set of Schur.

Set

For each there is a smallest, so that to exist for each color of a monochromatic solution.

Evidence

It should be. Ramsey's theorem ensures the existence of the number, for an arbitrary coloring of the complete graph with nodes, the existence of a monochromatic triangle. We select our coloring as follows. The nodes of the will with numbered and divided the amount into disjoint subsets. These quantities should correspond to the colors. Now the edge connecting the nodes and is dyed with the color of the amount to which it belongs. According to Ramsey's theorem there exists a monochromatic triangle in the graph and the corners are. Then follows since and are monochrome. Then applies with and. The theorem is proved.

Generalization

In addition the set of Rado a generalization can be achieved if, instead of the equation, the equation is considered.

Be and be for each. Then there exists a smallest number such that every coloring of at least one exists, that a solution to the color exists.

Specialization

For the original and for the generalized case, in each case to examine whether the presence of these numbers is present, an additional requirement that is first and in the generalized case. Especially in this area are limited upper and lower bounds have been studied so far.

Others

• The numbers are called Schur numbers.
• The numbers are called general Schur numbers.
• The triples that satisfy the above set of hot Schurtripel.
• The tuple of generalization called Schur t- tuples.
• The set of Rado represents a generalization of Schur's theorem

While the Schur's figures, the research focus relates to the determination of barriers, interested in the tuples, the number, so how many of the tuple for exist.