On the number of generalized Sidon sets

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A set A of nonnegative integers is called a Sidon set if there is no Sidon 4-tuple, i.e., (a, b, c, d) in A with a + b = c + d and {a, b} ∩ {c, d} = ∅. Cameron and Erdős proposed the problem of determining the number of Sidon sets in [n]. Results of Kohayakawa, Lee, Rödl and Samotij, and Saxton and Thomason have established that the number of Sidon sets is between 2(1.16+o(1))√n and 2(6.442+o(1))√n. An α-generalized Sidon set in [n] is a set with at most α Sidon 4-tuples. One way to extend the problem of Cameron and Erdős is to estimate the number of α-generalized Sidon sets in [n]. We show that the number of (n/ log4 n)-generalized Sidon sets in [n] with additional restrictions is 2Θ(√n) . In particular, the number of (n/ log5 n)-generalized Sidon sets in [n] is 2Θ(√n) . Our approach is based on some variants of the graph container method.

Original languageEnglish (US)
Pages (from-to)3-21
Number of pages19
JournalActa Scientiarum Mathematicarum
Issue number1
StatePublished - 2021


  • Generalized sidon set
  • The graph container method

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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