Erdős–Rado theorem

The Erdős–Rado theorem gives cardinal bounds guaranteeing large homogeneous subsets in colorings of finite subsets of sufficiently large sets. Its best-known form concerns pairs colored with finitely many colors.

Partition calculus: The study of unavoidable homogeneous configurations in colorings of sets, expressed through cardinal partition relations. The theorem is a central cardinal partition relation in this framework.
Cardinal number: A number that measures the size of a set, including infinite sets. The theorem compares sizes of infinite sets using cardinal numbers.
Paul Erdős: A Hungarian mathematician whose work spanned number theory, combinatorics, and set theory. He co-developed the theorem and the partition-calculus program.
Ramsey theory: The study of conditions that force order and homogeneous patterns within sufficiently large structures. The theorem provides an infinite-cardinal guarantee within Ramsey theory.
Cardinal arithmetic: The arithmetic of infinite cardinal numbers, including exponentiation and successor operations. Its bound uses cardinal exponentiation and a successor cardinal.
Ordinal number: A number describing the order type of a well-ordered set. Ordinal-indexed notation distinguishes the finite-subset sizes in partition relations.
Richard Rado: A German-born British mathematician known for work in combinatorics and set theory. He co-developed the theorem with Erdős.
Erdős–Rado arrow notation: A notation expressing that every coloring of specified finite subsets contains a homogeneous subset of a stated size. It records the theorem's guarantee compactly as a partition relation.
Homogeneous set: A subset whose elements or finite subsets all receive the same color under a given coloring. The theorem guarantees such subsets inside large colored sets.
Power set: The set of all subsets of a given set. Colorings of pairs are functions defined on part of a power set.

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