Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New bounds on the strength of some restrictions of Hindman's Theorem

Lorenzo Carlucci, Leszek Aleksander Kołodziejczyk, Francesco Lepore, Konrad Zdanowski

TL;DR

Upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem are proved, highlighting the role of a sparsity-like condition on the solution set, which is called apartness.

Abstract

We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors implies $\mathsf{ACA}_0$. An emerging {\em leitmotiv} is that the known lower bounds for Hindman's Theorem and for its restriction to sums of at most 2 elements are already valid for a number of restricted versions which have simple proofs and better computability- and proof-theoretic upper bounds than the known upper bound for the full version of the theorem. We highlight the role of a sparsity-like condition on the solution set, which we call apartness.

New bounds on the strength of some restrictions of Hindman's Theorem

TL;DR

Upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem are proved, highlighting the role of a sparsity-like condition on the solution set, which is called apartness.

Abstract

We prove upper and lower bounds on the effective content and logical strength for a variety of natural restrictions of Hindman's Finite Sums Theorem. For example, we show that Hindman's Theorem for sums of length at most 2 and 4 colors implies . An emerging {\em leitmotiv} is that the known lower bounds for Hindman's Theorem and for its restriction to sums of at most 2 elements are already valid for a number of restricted versions which have simple proofs and better computability- and proof-theoretic upper bounds than the known upper bound for the full version of the theorem. We highlight the role of a sparsity-like condition on the solution set, which we call apartness.

Paper Structure

This paper contains 11 sections, 16 theorems, 13 equations, 1 table.

Table of Contents

  1. Introduction and motivation
  2. Hindman's Theorem, apartness, and finite unions
  3. Bounded Hindman vs. Ramsey
  4. Sums of at most two terms
  5. Sums of exactly three terms, with apartness
  6. Bounded Hindman and Polarized Ramsey
  7. Other restrictions of Hindman's Theorem
  8. Weak Yet Strong Principles
  9. Increasing Polarized Hindman's Theorem
  10. Exactly Large Sums, with apartness
  11. Conclusion and some open questions

Key Result

Proposition 1

For each positive integers $t$ and $k$, $\mathsf{HT}_k$ and $\mathsf{HT}_k$ with $t$-apartness are equivalent over $\mathsf{RCA}_0$. The equivalence is witnessed by strong computable reductions.

Theorems & Definitions (39)

  • Definition 2.1: Hindman's Theorem with bounded-length sums
  • Definition 2.2: Apartness Condition
  • Proposition 1: Implicit in Hin:72
  • Definition 2.3: Finite Unions Theorem
  • Proposition 2
  • proof
  • Corollary 3
  • Lemma 4: $\mathsf{RCA}_0$
  • proof
  • Proposition 5
  • ...and 29 more