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Claim. Theorem 4.1 of P. Erdős and F. Galvin, Some Ramsey-type theorems, paged at Theorem 4.1 of the library's source card, with the partition its proof gives: for every $\varphi:\mathbb{N}\to\mathbb{N}$ there is a partition $\mathbb{N}=C_1\cup C_2$ (with $\varphi$ first taken strictly increasing, write $x=2^ry$ with $y$ odd; $x\in C_2$ if $y\ge\varphi(2^{r+1})$, else $x\in C_1$) such that, for every infinite sequence $x_1<x_2<\cdots$ of positive integers, if the sums of consecutive terms $\mathrm{CFS}(\{x_1,x_2,\ldots\})$ lie in one class, then that class is $C_2$ and $x_n>\varphi(n)$ for all $n$. Since $\mathrm{CFS}\subseteq\mathrm{FS}$, a sequence whose finite sums all lie in one class has $x_n>\varphi(n)$ for every $n$. With two colors, finite sums that do not contain all colors lie in one class, so with $\varphi=f$ no sequence with $a_n<f(n)$ for even one $n$ qualifies, and for $k=2$ no $f$ has the property Problem 948 asks for. The same holds for colorings of all integers and sequences that may begin with nonpositive terms: color the nonpositive integers arbitrarily and apply the theorem with $\varphi(m)=\max_{j\le2m}f(j)$ to the positive tail $x_m=a_{m_0+m}$, whose finite sums are among those of the whole sequence (one authored line).

Covers. The case $k=2$ (two colors) for every $f$, which is Erdős's original monochromatic question; Erdős's 1977 report ([Er77c], p. 57) credited the coloring to Galvin without proof. The case $k=1$ fails trivially, and the paper's Problem 4.2 records the case of three classes as unknown. Every number of colors is the accepted full claim of 2026.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. Refereed: P. Erdős and F. Galvin, Some Ramsey-type theorems, Discrete Math. 87 (1991), no. 3, 261--269 (received 3 January 1989). Reviewed: the site's curator, T. F. Bloom, credits Galvin's coloring with the negative answer for $k=2$ in the problem's commentary, on the page labeled SOLVED (last edited 5 July 2026).

Dating. The page is dated by the issue month, February 1991 in the Crossref record; the day is a placeholder.