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"Weak yet strong" restrictions of Hindman's Finite Sums Theorem

Lorenzo Carlucci

Abstract

We present a natural restriction of Hindman's Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman's Theorem with analogous properties.

"Weak yet strong" restrictions of Hindman's Finite Sums Theorem

Abstract

We present a natural restriction of Hindman's Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman's Theorem with analogous properties.

Paper Structure

This paper contains 7 sections, 8 theorems, 12 equations.

Table of Contents

  1. Introduction and Motivation
  2. A family of restrictions of the Finite Sums Theorem
  3. An example: the Hindman-Brauer Theorem
  4. A family of restrictions of the Finite Sums Theorem admitting simple proofs
  5. Computability-theoretic and proof-theoretic upper bounds
  6. A lower bound on the Hindman-Brauer Theorem
  7. Conclusions

Key Result

Theorem 1

For all $r,\ell,s\geq 1$ there exists $n=n(r,\ell,s)$ such that if $g:[1,n]\to r$ then there exists $a,b>0$ such that $\{a,a+b,a+2b,\dots,a+ (\ell-1)b\}\cup \{sb\}\subseteq [1,n]$ is monochromatic.

Theorems & Definitions (20)

  • Theorem 1: Brauer's Theorem, Bra:28
  • Definition 1: Apartness Condition
  • Theorem 2: Hindman-Brauer Theorem
  • proof
  • proof : Second proof of Theorem \ref{['thm:hinbrau']}
  • Theorem 3: Jockusch, Joc:72
  • Theorem 4: Jockusch, Joc:72
  • Proposition 1
  • Proposition 2
  • Theorem 5
  • ...and 10 more