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 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.
Part of this work was done while the first author was visiting the Institute for Mathematical Sciences, National University of Singapore in 2016. The visit was supported by the Institute. The second author was partially supported by Polish National Science Centre grant no. 2013/09/B/ST1/04390. The fourth author was partially supported by University Cardinal Stefan Wyszyński in Warsaw grant UmoPBM-26/16.
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Notes
- 1.
- 2.
The Finite Unions Theorem states that every coloring of the finite non-empty sets of \(\mathbf {N}\) admits an infinite and pairwise unmeshed family H of finite non-empty sets (sometimes called a block sequence) such that every finite non-empty union of elements of H is of the same color. Two finite non-empty subsets x, y of \(\mathbf {N}\) are unmeshed if either \(\max {x}<\min {y}\) or \(\max {y}<\min {x}\). Note that Hindman’s Theorem is equivalent to the Finite Unions Theorem only if the pairwise unmeshed condition is present.
- 3.
Note that the latter result is not present in the diagram in [11]. \(\mathsf {D}^2_2\), defined in [7], is the following assertion: For every 0, 1-valued function f(x, s) for which a \(\lim _{s\rightarrow \infty }f(x,s)\) exists for each x, there is an infinite set H and a \(k<2\) such that for all \(h\in H\) we have \(\lim _{s\rightarrow \infty }f(h,s)=k\).
- 4.
We thank Ludovic Patey for pointing out to us the results implying strictness.
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Carlucci, L., Kołodziejczyk, L.A., Lepore, F., Zdanowski, K. (2017). New Bounds on the Strength of Some Restrictions of Hindman’s Theorem. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_21
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