In every 2-coloring c: N→0,1, does there exist an infinite set A⊆ N such that all elements of its sumset A+A=a+b:a,b∈ A have the same color? Equivalently, must there exist an infinite A such that, for all n,m∈ A+A, c(n)=c(m)?

Erdős Problem #1199 — Monochromatic Sumsets in Two-Colorings

In every 2-coloring $c:\mathbb N\to\{0,1\}$ , does there exist an infinite set $A\subseteq\mathbb N$ such that all elements of its sumset

A+A=\{a+b:a,b\in A\}

have the same color? Equivalently, must there exist an infinite $A$ such that, for all $n,m\in A+A$ , $c(n)=c(m)$ ?

References

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