Английская Википедия:Erdős–Rado theorem

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after Paul Erdős and Richard Rado.Шаблон:Sfnp It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis,Шаблон:Sfnp and hence the result is sometimes also referred to as the Erdős–Rado–Kurepa theorem.

Statement of the theorem

If r ≥ 0 is finite and κ is an infinite cardinal, then

<math>

\exp_r(\kappa)^+\longrightarrow(\kappa^+)^{r+1}_\kappa </math> where exp0(κ) = κ and inductively expr+1(κ)=2expr(κ). This is sharp in the sense that expr(κ)+ cannot be replaced by expr(κ) on the left hand side.

The above partition symbol describes the following statement. If f is a coloring of the r+1-element subsets of a set of cardinality expr(κ)+, in κ many colors, then there is a homogeneous set of cardinality κ+ (a set, all whose r+1-element subsets get the same f-value).

Notes

Шаблон:Reflist

References

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Erdős–Rado_theorem&oldid=14560530