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Effectiveness of Hindman's Theorem for Bounded Sums
TLDR
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. Expand
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Three topological reducibilities for discontinuous functions
We define a family of three related reducibilities, $\leq_T$, $\leq_{tt}$ and $\leq_m$, for arbitrary functions $f,g:X\rightarrow\mathbb R$, where $X$ is a compact separable metric space. TheExpand
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A note on the diamond operator.
We show that if $1 \leq_W F$ and $F \star F \leq_W F$, then $F^\diamond \leq_W F$, where $\star$ and $\diamond$ are the following operations in the Weihrauch lattice: $\star$ is the compositionalExpand
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Ramsey's theorem for singletons and strong computable reducibility
We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducibleExpand
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Seas of squares with sizes from a $\Pi^0_1$ set
For each $\Pi^0_1$ $S\subseteq \mathbb{N}$, let the $S$-square shift be the two-dimensional subshift on the alphabet $\{0,1\}$ whose elements consist of squares of 1s of various sizes on a backgroundExpand
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Seas of squares with sizes from a Π10 set
For each Π10S ⊆ N, let the S-square shift be the two-dimensional subshift on the alphabet {0, 1} whose elements consist of squares of 1s of various sizes on a background of 0s, where the side lengthExpand
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A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK
  • L. Westrick
  • Mathematics, Computer Science
  • The Journal of Symbolic Logic
  • 13 February 2013
TLDR
We show that for each recursive ordinal $\alpha > 0$, the set of Turing indices of $C[0,1]$ functions that are differentiable with rank at most α is ${{\rm{\Pi }}_{2\alpha + 1}}$ -complete. Expand
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Effectiveness for the Dual Ramsey Theorem
We analyze the Dual Ramsey Theorem for $k$ partitions and $\ell$ colors ($\mathsf{DRT}^k_\ell$) in the context of reverse math, effective analysis, and strong reductions. Over $\mathsf{RCA}_0$, theExpand
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Finding bases of uncountable free abelian groups is usually difficult
We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending onExpand
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The reverse mathematics of Hindman's Theorem for sums of exactly two elements
TLDR
We show that there is a computable instance of HT$^{=2}_2$ such that all solutions can compute a function that is diagonally noncomputable relative to $\emptyset'$. Expand
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