Abstract

We show that for $n$ at least $10^{11}$, any 2-coloring of the $n$-dimensional grid $[4]^n$ contains a monochromatic combinatorial line. This is a special case of the Hales--Jewett theorem [Hales and Jewett, Trans. Amer. Math., 106 (1963), pp. 222--229], to which the best known general upper bound is due to Shelah [J. Amer. Math. Soc., 1 (1988), pp. 683--697]; Shelah's recursion gives an upper bound between $2 \uparrow \uparrow 7$ and $2 \uparrow \uparrow 8$ for the case we consider, and no better value was previously known.

Keywords

  1. Ramsey theory
  2. Hales--Jewett theorem
  3. coloring
  4. grid

MSC codes

  1. 05D10

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References

1.
N. Alon and J. Spencer, The Probabilistic Method, John Wiley & Sons, Hoboken, NJ, 2008.
2.
E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canad. Math. Bull., 11 (1968), pp. 409--414.
3.
V. Chvátal, Some unknown van der Waerden numbers, in Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 31--33.
4.
A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc., 106 (1963), pp. 222--229.
5.
N. Hindman and E. Tressler, The first nontrivial Hales--Jewett number is four, Ars Combin., 113 (2014), pp. 385--390.
6.
S. Shelah, Primitive recursive bounds for van der Waerden numbers, J. Amer. Math. Soc., 1 (1988), pp. 683--697.

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 1333 - 1342
ISSN (online): 1095-7146

History

Submitted: 13 April 2015
Accepted: 6 April 2016
Published online: 23 June 2016

Keywords

  1. Ramsey theory
  2. Hales--Jewett theorem
  3. coloring
  4. grid

MSC codes

  1. 05D10

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