Sidon sets are a special case of so-called sets. A set is called a set if for every the equation has at most different solutions with being elements of . The Sidon sets are sets.
Every finite set has a rather large subset. Komlós, Sulyok, and Szemerédi proved that in every -element set there is a subset of size at least for . For Sidon sets Abbott improved that to . It is not known whether one can take .
The infinite sets are understood even worse. Erdős [3, p. 89] proved that for every infinite Sidon set we have for some constant . On the other hand, for a long time no example of a set for which for some was known. Only recently Ruzsa used an extremely clever construction to prove the existence of a set for which for every and for all sufficiently large .
For an excellent survey of Sidon sets see .
- 1 Harvey Leslie Abbott. Sidon sets. Canad. Math. Bull., 33(3):335–341, 1990. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0715.11004Zbl 0715.11004.
- 2 Ben Green. sets: The current state of affairs. http://www.dpmms.cam.ac.uk/ bjg23/papers/bhgbounds.dvihttp://www.dpmms.cam.ac.uk/ bjg23/papers/bhgbounds.dvi, 2000.
- 3 Heini Halberstam and Klaus Friedrich Roth. Sequences. Springer-Verlag, second edition, 1983. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0498.10001Zbl 0498.10001.
- 4 János Komlós, Miklos Sulyok, and Endre Szemerédi. Linear problems in combinatorial number theory. Acta Math. Acad. Sci. Hung., 26:113–121, 1975. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0303.10058Zbl 0303.10058.
- 5 Kevin O’Bryant. A complete annotated bibliography of work related to Sidon sequences. Electronic Journal of Combinatorics. http://arxiv.org/abs/math.NT/0407117http://arxiv.org/abs/math.NT/0407117.
- 6 Imre Ruzsa. An infinite Sidon sequence. J. Number Theory, 68(1):63–71, 1998. Available at http://www.math-inst.hu/ ruzsa/cikkek.htmlhttp://www.math-inst.hu/ ruzsa/cikkek.html.
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