Hindman's theorem states that every finite coloring of the positive integers contains an infinite set 
whose nonempty finite subset sums
all have the same color (Hindman 1974). A member of 
may be used at most once in any one sum. Thus one color class
contains
 |
The conclusion uses a single infinite set, rather than a different finite set for each desired size. It is a theorem about sums alone. Requiring products as well gives a different question, Hindman's finite sums and products conjecture.
This is problem 532 in the Erdős problems collection (Bloom 2026).
