### Abstract

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^{≤n}_{k} denote the assertion that for each k-coloring c of ℕ there is an infinite set X ⊆ ℕ such that all sums Σ_{x}_{∈}_{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^{≤2}_{2}is not provable in RCA0 and in fact we show that it implies SRT^{2}_{2}in RCA_{0} +BΠ^{0}_{1}. We also show that there is a computable instance of HT^{≤3}_{3}with all solutions computing 0′. The proof of this result shows that HT^{≤3}_{3} implies ACA_{0} in RCA_{0}.

Original language | English (US) |
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Pages (from-to) | 134-142 |

Number of pages | 9 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 10010 |

DOIs | |

State | Published - Jan 1 2017 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*,

*10010*, 134-142. https://doi.org/10.1007/978-3-319-50062-1_11