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.

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

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 ${\displaystyle S=\{s_1,s_2,s_3,\dots \}}$ 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 ${\displaystyle S=p(\mathbb{\mathbb{N}})\cap \mathbb{\mathbb{N}}}$ where ${\displaystyle p}$ is a polynomial with ${\displaystyle p(0)=0}$ and positive leading coefficient, then ${\displaystyle S}$ 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.
• ${\displaystyle k\cdot \mathbb{\mathbb{N}}=\{k,2k,3k,\dots \}}$ is large.
• If S is large, ${\displaystyle k\cdot S}$ is also large.

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

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.

• Partition of a set

• 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. 

• Mathworld: van der Waerden's Theorem

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