Large set (Ramsey theory)

In Ramsey theory, a set S of natural numbers is considered to be a large set if and only if Van der Waerden's theorem can be generalized to assert the existence of arithmetic progressions with common difference in S. That is, S is large if and only if every finite partition of the natural numbers has a cell containing arbitrarily long arithmetic progressions having common differences in S.

Examples

 * The natural numbers are large. This is precisely the assertion of Van der Waerden's theorem.
 * The even numbers are large.

Properties
Necessary conditions for largeness include:
 * If S is large, for any natural number n, S must contain at least one multiple (equivalently, infinitely many multiples) of n.
 * If $$S=\{s_1,s_2,s_3,\dots\}$$ is large, it is not the case that sk≥3sk-1 for k≥ 2.

Two sufficient conditions are:
 * If S contains n-cubes for arbitrarily large n, then S is large.
 * If $$S =p(\mathbb{N}) \cap \mathbb{N}$$ where $$p$$ is a polynomial with $$p(0)=0$$ and positive leading coefficient, then $$S$$ is large.

The first sufficient condition implies that if S is a thick set, then S is large.

Other facts about large sets include: If $$S$$ is large, then for any $$m$$, $$S \cap \{ x : x \equiv 0\pmod{m} \}$$ is large.
 * If S is large and F is finite, then S – F is large.
 * $$k\cdot \mathbb{N}=\{k,2k,3k,\dots\}$$ is large.
 * If S is large, $$k\cdot S$$ is also large.

2-large and k-large sets
A set is k-large, for a natural number k > 0, when it meets the conditions for largeness when the restatement of van der Waerden's theorem is concerned only with k-colorings. 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&gt;1
 * k-largeness for all k implies largeness.

It is unknown whether there are 2-large sets that are not also large sets. Brown, Graham, and Landman (1999) conjecture that no such sets exists.