## Duke Mathematical Journal

### On a question of Erdős and Moser

#### Abstract

For two finite sets of real numbers $A$ and $B$, one says that $B$ is sum-free with respect to $A$ if the sum set $\{b+b'\mid b, b'\in B, b\neq b'\}$ is disjoint from $A$. Forty years ago, Erdőos and Moser posed the following question. Let $A$ be a set of $n$ real numbers. What is the size of the largest subset $B$ of $A$ which is sum-free with respect to $A$?

In this paper, we show that any set $A$ of $n$ real numbers contains a set $B$ of cardinality at least $g(n) \ln n$ which is sum-free with respect to $A$, where $g(n)$ tends to infinity with $n$. This improves earlier bounds of Klarner, Choi, and Ruzsa and is the first superlogarithmic bound for this problem.

Our proof combines tools from graph theory together with several fundamental results in additive number theory such as Freiman's inverse theorem, the Balog-Szemerédi theorem, and Szemerédi's result on long arithmetic progressions. In fact, in order to obtain an explicit bound on $g(n)$, we use the recent versions of these results, obtained by Chang and by Gowers, where significant quantitative improvements have been achieved.

#### Article information

Source
Duke Math. J. Volume 129, Number 1 (2005), 129-155.

Dates
First available in Project Euclid: 15 July 2005

http://projecteuclid.org/euclid.dmj/1121448866

Digital Object Identifier
doi:10.1215/S0012-7094-04-12915-X

Mathematical Reviews number (MathSciNet)
MR2155059

#### Citation

Sudakov, B.; Szemerédi, E.; Vu, V. H. On a question of Erdős and Moser. Duke Math. J. 129 (2005), no. 1, 129--155. doi:10.1215/S0012-7094-04-12915-X. http://projecteuclid.org/euclid.dmj/1121448866.

#### References

• A. Balog and E. Szemerédi, A statistical theorem of set addition, Combinatorica 14 (1994), 263--268.
• A. Baltz, T. Schoen, and A. Srivastav, Probabilistic construction of small strongly sum-free sets via large Sidon sets, Colloq. Math. 86 (2000), 171--176.
• M.-C. Chang, A polynomial bound in Freiman's theorem, Duke Math. J. 113 (2002), 399--419.
• S. L. G. Choi, On a combinatorial problem in number theory, Proc. London Math. Soc. (3) 23 (1971), 629--642.
• P. Erdős, Extremal problems in number theory'' in Proceedings of Symposia in Pure Mathematics, Vol. VIII, Amer. Math. Soc., Providence, 1965, 181--189.
• P. Erdős and E. Szemerédi, On sums and products of integers'' in Studies in Pure Mathematics, Birkhäuser, Basel, 1983, 213--218.
• G. A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr. 37, Amer. Math. Soc., Providence, 1973.
• W. T. Gowers, A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529--551.
• --. --. --. --., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001), 465--588.; Erratum, Geom Funct. Anal. 11 (2001), 869. ;
• R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Problem Books in Math., Unsolved Probl. in Intuitive Math. 1, Springer, New York, 1994.
• M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Grad. Texts in Math. 165, Springer, New York, 1996.
• I. Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 (1994), 379--388.
• --------, Sum-avoiding subsets, to appear in Ramanujan J.
• E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199--245.
• E. Szemerédi and V. H. Vu, Finite and infinite arithmetic progressions in sumsets, to appear in Ann. of Math. (2), preprint, 2003, http://www.math.ucsd.edu/$\tilde\ \,$vanvu/papers.html