The large sets conjecture

Importance: High ✭✭✭
Recomm. for undergrads: no
Posted by: vjungic
on: June 10th, 2007

\begin{conjecture} If $A$ is 2-large, then $A$ is large. \end{conjecture}

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/} For $r\in \mathbb{N}$, a set of positive integers $L$ is said to be $r$-large if for any $r$-coloring $f$ of positive integers there are arbitrarily long $f$ - monochromatic arithmetic progressions whose common differences belong to $L$. Then $L$ is large if and only if it is $r$-large for all $r$. From Bergelson-Leibman's Polynomial van der Waerden's Theorem \cite{BL} it follows that $\{ |p(n)| : n \in \mathbb{N} \} \cap \mathbb{N}$ is large for any polynomal $p$ with rational coefficients and such that $p(0)=0$.

The conjecture was stated in 1995 and published in 1999 \cite{BGL}.

Bibliography

% Example:

[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, \href[On the set of common differences in van der Waerden’s theorem on arithmetic progressions]{http://www.math.ucsd.edu/~fan/ron/papers/99_02_common_differences.pdf}, 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.

%*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)


* indicates original appearance(s) of problem.