Large set (Ramsey theory)
# 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 $s_k≥3s_{k-1}$ 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 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.

If $S$ is large, then for any $m$, $S\cap \{x:x\equiv 0{\pmod {m}}\}$ is 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>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.

