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1. Borel combinatorics fail in HYP

   https://doi.org/10.1142/S0219061322500234  

   Towsner, Henry; Weisshaar, Rose; Westrick, Linda (August 2023, Journal of Mathematical Logic)

   We characterize the completely determined Borel subsets of HYP as exactly the [Formula: see text] subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. 

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2. Equivalence of codes for countable sets of reals

   https://doi.org/10.4153/S0008439520000661  

   Chan, William (September 2021, Canadian Mathematical Bulletin)

   null (Ed.)

   Abstract A set $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ is universal for countable subsets of $${\mathbb {R}}$$ if and only if for all $$x \in {\mathbb {R}}$$ , the section $$U_x = \{y \in {\mathbb {R}} : U(x,y)\}$$ is countable and for all countable sets $$A \subseteq {\mathbb {R}}$$ , there is an $$x \in {\mathbb {R}}$$ so that $$U_x = A$$ . Define the equivalence relation $$E_U$$ on $${\mathbb {R}}$$ by $$x_0 \ E_U \ x_1$$ if and only if $$U_{x_0} = U_{x_1}$$ , which is the equivalence of codes for countable sets of reals according to U . The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $$E_{U^*}$$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $$E_U$$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $$E_U$$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $$\mathbf {\Sigma }_3^1$$ -generic absoluteness, then for all $$U \subseteq {\mathbb {R}} \times {\mathbb {R}}$$ that are $$\mathbf {\Sigma }_1^1$$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $$E_U$$ . 

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3. Orienting Borel graphs

   https://doi.org/10.1090/proc/15742  

   Thornton, Riley (April 2022, Proceedings of the American Mathematical Society)

   We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by k k for various cardinals k k . We show that for a probability measure preserving (p.m.p) graph G G , a measurable orientation can be found when k k is larger than the normalized cost of the restriction of G G to any positive measure subset. Using an idea of Conley and Tamuz, we can also find Borel orientations of graphs with subexponential growth; however, for every k k we also find graphs which admit measurable orientations with outdegree bounded by k k but no such Borel orientations. Finally, for special values of k k we bound the projective complexity of Borel k k -orientability for graphs and graphings of equivalence relations. It follows from these bounds that the set of equivalence relations admitting a Borel selector is Σ 2 1 \mathbf {\Sigma }_{2}^{1} in the codes, in stark contrast to the case of smooth relations. 

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4. Simulations and symmetries

   https://doi.org/10.1093/mnras/staa251  

   Modi, Chirag; Chen, Shi-Fan; White, Martin (March 2020, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT We investigate the range of applicability of a model for the real-space power spectrum based on N-body dynamics and a (quadratic) Lagrangian bias expansion. This combination uses the highly accurate particle displacements that can be efficiently achieved by modern N-body methods with a symmetries-based bias expansion which describes the clustering of any tracer on large scales. We show that at low redshifts, and for moderately biased tracers, the substitution of N-body-determined dynamics improves over an equivalent model using perturbation theory by more than a factor of two in scale, while at high redshifts and for highly biased tracers the gains are more modest. This hybrid approach lends itself well to emulation. By removing the need to identify haloes and subhaloes, and by not requiring any galaxy-formation-related parameters to be included, the emulation task is significantly simplified at the cost of modelling a more limited range in scale. 

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5. Katok’s special representation theorem for multidimensional Borel flows

   https://doi.org/10.1017/etds.2023.62  

   SLUTSKY, KONSTANTIN (October 2023, Ergodic Theory and Dynamical Systems)

   Katok’s special representation theorem states that any free ergodic measure-preserving R^d-flow can be realized as a special flow over a Z^d-action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel R^d-flows emerge from Z^d-actions through the special flow construction using bi-Lipschitz cocycles. 

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