On the number of generalized Sidon sets

TL;DR

This paper extends the study of Sidon sets by estimating the number of generalized Sidon sets with bounded Sidon 4-tuples in [n], using graph container methods to establish exponential bounds in terms of f3f3n.

Contribution

It provides bounds on the number of lpha-generalized Sidon sets in [n], extending previous results and introducing new techniques based on graph container variants.

Findings

Number of (n/log^4 n)-generalized Sidon sets is 2^{f3f3n(f3f3n)}.

Number of (n/log^5 n)-generalized Sidon sets is 2^{f3f3n(f3f3n)}.

Method relies on variants of the graph container technique.

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 Erd\H os proposed the problem of determining the number of Sidon sets in $[n]$. Results of Kohayakawa, Lee, R\" odl and Samotij, and Saxton and Thomason has 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 Erd\H os 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.