# A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

@article{Carlucci2018AWV,
title={A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem},
author={Lorenzo Carlucci},
journal={Archive for Mathematical Logic},
year={2018},
volume={57},
pages={381-389}
}
• L. Carlucci
• Published 18 October 2016
• Mathematics
• Archive for Mathematical Logic
Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over $$\mathbf {RCA}_0$$RCA0 to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey…
Results of Hindman, Leader and Strauss and of the second author and Rinot showed that some natural analogs of Hindman’s theorem fail for all uncountable cardinals. Results in the positive direction
“Weak yet strong” restrictions of Hindman’s Finite Sums Theorem
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
An adjacent Hindman theorem for uncountable groups
• Mathematics
• 2020
Results of Hindman, Leader and Strauss and of the second author and Rinot showed that some natural analogs of Hindman's theorem fail for all uncountable cardinals. Results in the positive direction
New Bounds on the Strength of Some Restrictions of Hindman's Theorem
• Mathematics
CiE
• 2017
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.
A Note on Hindman-Type Theorems for Uncountable Cardinals
It is shown how a family of natural Hindman-type theorems for uncountable cardinals can be obtained by adapting some recent results of the author from their original countable setting.
Hindman’s theorem for sums along the full binary tree, $$\Sigma ^0_2$$-induction and the Pigeonhole principle for trees
• Mathematics
Archive for Mathematical Logic
• 2022
<jats:p>We formulate a restriction of Hindman’s Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the

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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
New Bounds on the Strength of Some Restrictions of Hindman's Theorem
• Mathematics
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• 2017
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.
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