Introduction

Erdos Basis Problem

Let $k\in \mathbb{N}\cup \{0\}$ and let $A\subset \mathbb{N}$ be a set of non-negative integers such that for every $n\in \mathbb{N}$ there exist $a,b\in A$ such that $a+b=n$. We call $A$ a basis of order 2 of the natural number. Does it follow that there exists $n\in \mathbb{N}$ such that the number of ways of writing $n$ as $a+b$ with $a,b\in A$ is at least $k$?

notation: $A+A=\{x+y:x,y\in A\}$

Definition

A set $X\subset \mathbb{N}$ is a sidon set if the only quadruples $(a,b,c,d)\in A^4$ such that $a+b=c+d$ are the trivial ones where $\{a,b\}=\{c,d\}$.