Effectiveness of hindman’s theorem for bounded sums

Damir D. Dzhafarov, Carl G. Jockusch, Reed Solomon, Linda Brown Westrick

Research output: Contribution to journalArticle

5 Scopus citations

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≤nk denote the assertion that for each k-coloring c of ℕ there is an infinite set X ⊆ ℕ such that all sums ΣxF 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.

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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