We consider the strength and effective content of restricted versions of Hindman’s Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let HT≤nk denote the assertion that for each k-coloring c of ℕ there is an infinite set X ⊆ ℕ such that all sums Σx∈F x for F ⊆ X and 0 < |F| ≤ n have the same color. We prove that there is a computable 2-coloring c of ℕ such that there is no infinite computable set X such that all nonempty sums of at most 2 elements of X have the same color. It follows that HT≤22is not provable in RCA0 and in fact we show that it implies SRT22in RCA0 +BΠ01. We also show that there is a computable instance of HT≤33with all solutions computing 0′. The proof of this result shows that HT≤33 implies ACA0 in RCA0.
|Original language||English (US)|
|Number of pages||9|
|Journal||Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)|
|State||Published - 2017|
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)