Erdős Problem #948

The problem

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?

Statement quoted from erdosproblems.com/948, maintained by Thomas Bloom. Original source: [Er77c,p.57][ErGa91,p.268]. Current status there: solved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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