Erdős–Rado theorem

The set of Erdős - Rado, named after Paul Erdős and Richard Rado, is a mathematical theorem in the field of set theory. He can make a statement about how big does a lot to have a certain decomposition property.

The arrow notation

If the following statement is true:

This goes back to P. Erdős and R. Rado arrow notation will be illustrated here by some examples. The case simply means that in a decomposition into parts of at least one part must have the cardinality. Of cardinality reasons alone, ie valid for infinite cardinal numbers, or that, with the aleph notation for the smallest infinite cardinal number is. More interesting, that is less trivial statements are only obtained for the case. This allows to formulate the set of Ramsey in the arrow notation as follows:

Note that the statement remains true if one goes to larger cardinal numbers or if you shrink one of the sizes. In the arrow notation, the variables are therefore separated by the arrow according to their monotonicity behavior, which is at least a hint.

Does not the statement, we write also. If, as some authors leave the index off you.

W. Sierpiński has shown for infinite cardinal numbers that

The successor cardinal number of designated.

Wording of the sentence

Using the above arrow notation and Beth function of the set of Erdos - Rado is:

• For all natural numbers.

For is and the set of Erdos - Rado only states that is at a decomposition of into countably many parts must at least have a part of the thickness, and that means that not countable union of countable sets is. Only for one obtains non-trivial statements.

The above goes back to Sierpiński statement said for that, or because of the monotony behavior for all. The set of Erdos - Rado now makes the positive statement for all, because for obtained and the monotony behavior of the arrow notation leads to the desired statement.

The above theorem allows the following generalization to higher thicknesses, which is also referred to as a set of Erdos - Rado. For an infinite cardinal number defining recursively

Then we have

• For all natural numbers and all infinite cardinal numbers.

This result is sharp, that is the cardinal number on the left side of the arrow can not be replaced by a smaller one. Therefore, the set of Erdos - Rado is a statement about how great a cardinal number must be to ensure that the partition property is fulfilled: There must be.

For, and the aforementioned set of Erdos - Rado is obtained as a special case. From the monotonicity properties of the arrow notation follows from the case of the set of Erdos - Rado: