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We provide analogues of the results from Friedman and Motto Ros (2011) and Camerlo, Marcone, and Motto Ros (2013) (which correspond to the case

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27-29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s.The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters.This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas.

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

The current paper proves the results announced in [5]. We isolate a new large cardinal concept, \"remarkability.\" Consistencywise, remarkable cardinals are between ineffable and ω-Erdos cardinals. They are characterized by the existence of \"O#-like\" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness for proper forcings. In particular, said absoluteness does not imply Π11 determinacy.

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k ≥ 1 and A, B ⊆ ℝᵏ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of Zᵈ.

We show that if all collections of infinite subsets of ℕ have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E₀ and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.

We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\\sigma $ . For $p\\in \\left [1,2\\right )\\cup \\left (2,\\infty \\right )$ , we show that the isometry classes of $L_p[0,1]$ and $\\ell _p$ are $G_\\delta $ -complete sets and $F_{\\sigma \\delta }$ -complete sets, respectively. Then we show that the isometry class of $c_0$ is an $F_{\\sigma \\delta }$ -complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces; for instance, the class of separable $\\mathcal {L}_{p,\\lambda +}$ -spaces, for $p,\\lambda \\geq 1$ , is shown to be a $G_\\delta $ -set, the class of superreflexive spaces is shown to be an $F_{\\sigma \\delta }$ -set, and the class of spaces with local $\\Pi $ -basis structure is shown to be a $\\boldsymbol {\\Sigma }^0_6$ -set. The paper is concluded with many open problems and suggestions for a future research.

We study the coarse classification of partial orderings using chain conditions in the context of descriptive combinatorics. We show that (unlike the Borel counterpart of many other combinatorial notions), we have a strict hierarchy of different chain conditions, similar to the classical case.

The author develops the theory of Hod mice below AD_{\\mathbb{R}}+ \"\\Theta is regular\". He uses this theory to show that HOD of the minimal model of AD_{\\mathbb{R}}+ \"\\Theta is regular\" satisfies GCH. Moreover, he shows that the Mouse Set Conjecture is true in the minimal model of AD_{\\mathbb{R}}+ \"\\Theta is regular\".