Van der Waerden's theorem

For every finite coloring of the positive integers and every positive integer length, some color class contains an arithmetic progression of that length.

Van der Waerden number: The least integer such that every coloring of an interval of that size with a fixed number of colors contains a monochromatic arithmetic progression of a fixed length. These finite thresholds quantify the theorem’s guarantee for each choice of colors and progression length.
Bartel Leendert van der Waerden: A Dutch mathematician known for work in algebra and for the theorem bearing his name. He proved the theorem in response to a problem posed by Emil Artin and Otto Schreier.
Szemerédi's theorem: A theorem stating that every subset of the integers with positive upper density contains arbitrarily long arithmetic progressions. It strengthens coloring-based existence into a density-based guarantee.
Schur's theorem: A theorem stating that every finite coloring of the positive integers has a monochromatic solution to x + y = z. It forces one additive equation, whereas Van der Waerden guarantees progressions of every finite length.
Color-focusing argument: A combinatorial proof technique that repeatedly restricts attention to nested sets whose elements share a color pattern. The original proof builds a monochromatic progression by focusing colors across structured finite sets.
Emil Artin: A mathematician whose work shaped modern algebra and number theory. His question about monochromatic progressions prompted the result.
Hales–Jewett theorem: A theorem guaranteeing combinatorial lines in sufficiently high-dimensional words under finite colorings. Its proof yields a route to Van der Waerden’s theorem through structured words.
Ramsey's theorem: A family of theorems guaranteeing monochromatic substructures in sufficiently large colored structures. It shares the color-forced structure theme but centers on homogeneous subsets and subgraphs.
Induction: A proof method that establishes a base case and then derives each case from an earlier one. Induction on progression length underlies a classic proof of the finite theorem.
Otto Schreier: A mathematician known for contributions to group theory and topology. He joined Artin in posing the coloring problem that led to the theorem.

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