Erdős Problem #948 — 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…

Erdős and Galvin asked whether one can choose a function $f$ and a finite number of colours $k$ forcing a slowly growing sequence whose finite subset sums omit a colour.

Known results

• Galvin ruled out the monochromatic version for $k=2$ using a two-colouring based on powers of $2$.
• Erdős and Galvin showed that, for every $kge 2$, sequences with $a_n<2^{2^{O(n)}}$ can have finite interval sums using only two colours.
• The corresponding question for countably many colours was recorded as open.

Claimed negative answer (date not stated)

GPT Pro, prompted by Price, claimed that for every $f$ there is a colouring of $\mathbb{N}$ by $\mathbb{N}$ such that every sequence satisfying $a_n<f(n)$ for infinitely many $n$ has finite subset sums of every colour. A discussion gives a rough binary-digit interval construction, but no formal verification or independent publication was found.

Current status (as of April 2026): The GPT Pro construction is claimed to settle the problem negatively, while the proposed proof remains unverified and the problem is not resolved in the published record.