# The large sets conjecture

**Conjecture**If is 2-large, then is large.

For , a set of positive integers is said to be -large if for any -coloring of positive integers there are arbitrarily long - monochromatic arithmetic progressions whose common differences belong to . Then is large if and only if it is -large for all . From Bergelson-Leibman's Polynomial van der Waerden's Theorem [BL] it follows that is large for any polynomal with rational coefficients and such that .

The conjecture was stated in 1995 and published in 1999 [BGL].

## Bibliography

[BL] V. Bergelson and A. Leibman, Polynomial extension of van der Waerden’s and Szemer\'{e}di’s theorems, J. Amer. Math. Soc. 9 (1996) 725-753.

*[BGL] T.C. Brown, R. L. Graham, and B. M. Landman, On the set of common differences in van der Waerden’s theorem on arithmetic progressions, Canadian Math. Bull. 42 (1999) 25-36.

[J] V. Jungic, On Brown’s conjecture on Accessible Sets, J. Comb. Theory, Ser. A 110(1) (2005), 175-178

[LR] B. M. Landman and A. Robertson, Ramsey Theory on the Integers, American Mathematical Society, 2004.

* indicates original appearance(s) of problem.