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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.



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 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 where is a polynomial with and positive leading coefficient, then 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  F is large.
  • is large.
  • If S is large, is also large.

If is large, then for any , 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.

See also

Further reading

  • 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