2.1 Statement and proof of Ramsey's Theorem for infinite sets

Ramsey's Theorem for infinite sets is a powerful result in combinatorics. It states that for any r-coloring of pairs from an infinite set, there's always an infinite subset where all pairs share the same color.

This theorem extends finite Ramsey theory to infinite cases, providing a foundation for further study. It applies to both countable and uncountable sets, showcasing the far-reaching implications of Ramsey-type results in mathematics.

Ramsey's Theorem for Infinite Sets

Ramsey's Theorem for infinite sets

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• Ramsey's Theorem for infinite sets states for any infinite set X and positive integer r, any r-coloring of X's 2-element subsets yields an infinite subset Y of X where all 2-element subsets of Y share the same color
• Applies to countably infinite sets (natural numbers) and uncountably infinite sets (real numbers)
• Generalizes finite Ramsey's Theorem to infinite case, providing foundation for infinite Ramsey theory
• Guarantees existence of infinite homogeneous subsets for any r-coloring, not limited to 2-colorings

Components of Ramsey's Theorem

• Infinite set X serves as starting point (natural numbers, real numbers)
• r-coloring assigns colors to all 2-element subsets of X
• Infinite monochromatic subset Y exists within X
• Y exhibits homogeneity all its 2-element subsets share same color
• Theorem holds for any number of colors r, showcasing robustness

Proof of Ramsey's Theorem

1. Use infinite Ramsey's Theorem for pairs as foundation
2. Apply induction on number of colors r
3. Establish base case: r = 1 (trivial, all subsets monochromatic)
4. Inductive step: Assume theorem holds for r-1 colors
5. Consider r-coloring of 2-element subsets of X
6. Construct sequence of elements and subsets
7. Apply König's Lemma to infinite tree
8. Obtain infinite path in tree
9. Conclude existence of infinite monochromatic subset

• Proof employs key techniques induction, König's Lemma, diagonal argument
• Alternative approach uses Zorn's Lemma

Examples of Ramsey's Theorem

• 2-coloring of pairs of natural numbers: Color (a, b) red if a < b, blue otherwise Infinite monochromatic subset emerges as increasing sequence
• Graph theory: Infinite complete graph with edges colored red or blue Infinite monochromatic complete subgraph guaranteed to exist
• Set theory: Coloring pairs from countably infinite set Infinite subset with all pairs sharing same color always present
• Number theory: Van der Waerden's theorem (infinite version) Existence of arbitrarily long monochromatic arithmetic progressions in any coloring of integers
• Applications span combinatorics, set theory, logic, theoretical computer science