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\}\cap \{c, d\}=\emptyset $. 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 $2^{(1.16+o(1))\sqrt {n}}$ and $2^{(6.442+o(1))\sqrt {n}}$. An $\alpha $-generalized Sidon set in $[n]$ is a set with at most $\alpha $ Sidon 4-tuples. One way to extend the problem of Cameron and Erdős is to estimate the number of $\alpha $-generalized Sidon sets in $[n]$. We show that the number of $(n/\log ^4 n)$-generalized Sidon sets in $[n]$ with additional restrictions is $2^{\Theta (\sqrt {n})}$. In particular, the number of $(n/\log ^5 n)$-generalized Sidon sets in $[n]$ is $2^{\Theta (\sqrt {n})}$. Our approach is based on some variants of the graph container method.

DOI: 10.14232/actasm-018-777-z

AMS Subject Classification (1991): 05A16, 05D05

Keyword(s): the graph container method, generalized Sidon set

received 1.3.2018, revised 9.8.2020, accepted 29.10.2020. (Registered under 27/2018.)