Abstract
We give simple randomized algorithms leading to new upper bounds for combinatorial problems of Choi and Erdös: For an arbitrary additive group G let P n (G) denote the set of all subsets S of G with n elements having the property that 0 is not in S + S. Call a subset A of G admissible with respect to a set S from P n (G) if the sum of each pair of distinct elements of A lies outside S. For S ∈ P n (G) let h(S) denote the maximal cardinality of a subset of S admissible with respect to S. In particular we show \(h(n) := \mathrm{min}\{h(S) \vert G \mathrm{group}, S \in P_n(G)\} = \mathcal{O}((\ln n)^2)\). The methodical innovation of the whole approach is the use of large Sidon sets.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Alon, N., Aronov, B., Suri, S.: Can visibility graphs be represented compactly? Discrete and Computational Geometry 12, 347–365 (1994)
Alon, N., Orlitsky, A.: Repeated communication and Ramsey graphs. IEEE Transactions on Information Theory 41, 1276–1289 (1995)
Choi, S.L.G.: On a combinatorial problem in number theory. Proc. London Math. Soc. 23(3), 629–642 (1971)
Erdös, P.: Extremal problems in number theory. In: Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, R.I. vol. 8, pp. 181–189 (1965)
Guy, R.F.: Unsolved problems in number theory, pp. 128–129. Springer, New York (1994)
Komlós, J., Sulyok, M., Szemerédi, E.: Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hungar. 26, 113–121 (1975)
Luczak, T., Schoen, T.: On strongly sum-free subsets of abelian groups. Coll. Math. 71, 149–151 (1996)
Thiele, T.: Geometric Selection Problems and Hypergraphs, PhD thesis, Freie Universität Berlin (October 1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Baltz, A., Schoen, T., Srivastav, A. (1999). Probabilistic Construction of Small Strongly Sum-Free Sets via Large Sidon Sets. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-48413-4_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66329-4
Online ISBN: 978-3-540-48413-4
eBook Packages: Springer Book Archive