Large set (Ramsey theory)

Sets big enough to assert the existence of arithmetic progressions with common difference

For other uses, see Large set (disambiguation).

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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

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

## Properties

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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 , … } {\displaystyle 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 ( N ) ∩ N {\displaystyle S=p(\mathbb {N} )\cap \mathbb {N} } where  p {\displaystyle p} is a polynomial with  p ( 0 ) = 0 {\displaystyle p(0)=0} and positive leading coefficient, then  S {\displaystyle 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 ⋅ N = { k , 2 k , 3 k , … } {\displaystyle k\cdot \mathbb {N} =\\{k,2k,3k,\dots \\}} is large.
* If S is large,  k ⋅ S {\displaystyle k\cdot S} is also large.

If  S {\displaystyle S} is large, then for any  m {\displaystyle m} ,  S ∩ { x : x ≡ 0 ( mod m ) } {\displaystyle S\cap \\{x:x\equiv 0{\pmod {m}}\\}} is large.

## 2-large and k-large sets

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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.

## See also

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* Partition of a set

## Further reading

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* Brown, Tom; Graham, Ronald; Landman, Bruce (1999). "On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions". Canadian Mathematical Bulletin. 42 (1): 25–36. doi:10.4153/cmb-1999-003-9.

## External links

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* Mathworld: van der Waerden's Theorem