Theorem 2.11 (Hindman’s Theorem). Assuming that:

$ℕ$ is finitely coloured

Then there exists infinitely many

{(x_{n})}_{n \geq 1}

such that

FS (X_{1}) : = {\sum_{i \in I} x_{i} : \emptyset \neq I \subseteq ℕ, I finite}

is monochromatic.