Abstract
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} = ø;. Cameron and Erdős proposed the problem of determining the number of Sidon sets in [n]. Results of Kohayakawa, Lee, Rodl and Samotij, and Saxton and Thomason have established that the number of Sidon sets is between \({2^{\left( {1.16 + o\left( 1 \right)} \right)\sqrt n }}\) and \({2^{\left( {6.442 + o\left( 1 \right)} \right)\sqrt n }}\). 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/ log4n)-generalized Sidon sets in [n] with additional restrictions is \({2^{\Theta \left( {\sqrt n } \right)}}\). In particular, the number of (n /log5n)-generalized Sidon sets in [n] is \({2^{\Theta \left( {\sqrt n } \right)}}\). Our approach is based on some variants of the graph container method.
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Communicated by P. Hajnal
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Balogh, J., Li, L. & Hajnal, P. On the number of generalized Sidon sets. ActaSci.Math. 87, 3–21 (2021). https://doi.org/10.14232/actasm-018-777-z
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DOI: https://doi.org/10.14232/actasm-018-777-z