Erdős Problem #948

declared status 'open'. Formalized: no. Is there a function $f(n)$ and a $k$ such that in any $k$-colouring of the integers there exists a sequence $a_1<\cdots$ such that $a_n<f(n)$ for infinitely many $n$ and the set$$\left\{ \sum_{i\in S}a_i : \textrm{finite }S\right\}$$does not contain all colours? Current best: Erd\H{o}s initially asked whether this is possible with the set being monochromatic, but Galvin showed that this is not always possible, considering the two colouring where, writing $n=2^km$ with $m$ odd, we colour $n$ red if $m\geq F(k)$ and blue if $m<F(k)$ (for some sufficiently quickly growing $F$). Prize: no. Tags: number theory, ramsey theory.

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