2-large
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In the field of Ramsey theory, a branch of combinatorial mathematics, a set is 2-large when it meets the conditions for largeness when the restatement of van der Waerden's theorem is concerned only with 2-colorings. A similar definition defines k-largeness for any k. All nonempty subsets of the set of natural numbers are trivially 1-large. Every set is either large or k-large for some maximal k. This follows from two important, albeit trivially true, facts:
- k-largeness implies (k-1)-largeness for k>1
- k-largeness for all k implies largeness.
It is unknown whether there are 2-large sets that are not also large. Brown, Graham, and Landman conjecture (see link below) that no such set exists. A proof that 2-largeness implies 3-largeness would likely generalize into an inductive proof to verify this conjecture, while a set that is p-large but not q-large (for p < q) would be a suitable counter-example.
