Oscillating subalgebras of the atomless countable Boolean algebra
We show that the big Ramsey degree of the Boolean algebra with 3 atoms within the countable atomless Boolean algebra is infinite.
math.LO↗
arXiv subjects
Spencer Unger
Publications and source records attributed to Spencer Unger.
$G_δ$ Circle Squaring
We show that a circle and square of the same area in $\mathbb{R}^2$ are equidecomposable by translations using $\mathbfΔ^0_2$ pieces. That is, pieces which are simultaneously $F_σ$ and $G_δ$ sets. This improves a result of Máthé-Noel-Pikhurko and is the best possible complexity in terms of the Borel hierarchy. More generally we show that bounded sets $A,B \subseteq \mathbb{R}^n$ with small enough boundaries and the same nonzero Lebesgue measure are equidecomposable with pieces that are countable unions of finite Boolean combinations of translates of $A,B$, and open sets. The improvement comes from constructions of low complexity toasts and related objects which should be independently useful within Borel combinatorics.
math.LO↗
The tree property on long intervals of regular cardinals
In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.
math.LO↗
Stationary Reflection and the failure of SCH
In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal $ν$ such that the singular cardinal hypothesis fails at $ν$ and every collection of fewer than $\mathrm{cf}(ν)$ stationary subsets of $ν^+$ reflects simultaneously. For $\mathrm{cf}(ν) > ω$, this situation was not previously known to be consistent. Using different methods, we reduce the upper bound on the consistency strength of this situation for $\mathrm{cf}(ν) = ω$ to below a single partially supercompact cardinal. The previous upper bound of infinitely many supercompact cardinals was due to Sharon.
math.LO↗
Borel factors and embeddings of systems in subshifts
In this paper we study the combinatorics of free Borel actions of the group $\mathbb Z^d$ on Polish spaces. Building upon recent work by Chandgotia and Meyerovitch, we introduce property $F$ on $\mathbb Z^d$-shift spaces $X$ under which there is an equivariant map from any free Borel action to the free part of $X$. Under further entropic assumptions, we prove that any subshift $Y$ (modulo the periodic points) can be Borel embedded into $X$. Several examples satisfy property $F$ including, but not limited to, the space of proper $3$-colourings, tilings by rectangles (under a natural arithmetic condition), proper $2d$-edge colourings of $\mathbb Z^d$ and the space of bi-infinite Hamiltonian paths. This answers questions raised by Seward, and Gao-Jackson, and recovers a result by Weilacher and some results announced by Gao-Jackson-Krohne-Seward.
math.DS↗
Diagonal supercompact Radin forcing
Motivated by the goal of constructing a model in which there are no $κ$-Aronszajn trees for any regular $κ>\aleph_1$, we produce a model with many singular cardinals where both the singular cardinals hypothesis and weak square fail.
math.LO↗
Successive failures of approachability
Motivated by showing that in ZFC we cannot construct a special Aronszajn tree on some cardinal greater than $\aleph_1$, we produce a model in which the approachability property fails (hence there are no special Aronszajn trees) at all regular cardinals in the interval $[\aleph_2, \aleph_{ω^2+3}]$ and $\aleph_{ω^2}$ is strong limit.
math.LO↗
Stationary Reflection
We improve the upper bound for the consistency strength of stationary reflection at successors of singular cardinals.
math.LO↗
The strong tree property and weak square
We show that it is consistent, relative to $ω$ many supercompact cardinals, that the super tree property holds at $\aleph_n$ for all $2 \leq n < ω$ but there are weak square and a very good scale at $\aleph_ω$.
math.LO↗
Large Cardinal Axioms from Tameness in AECs
We show that various tameness assertions about abstract elementary classes imply the existence of large cardinals under mild cardinal arithmetic assumptions.
math.LO↗
Baire measurable paradoxical decompositions via matchings
We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann-Day problem: if $a$ is a nonamenable action of a group on a Polish space $X$ by Borel automorphisms, then there is a free Baire measurable action of $\mathbb{F}_2$ on $X$ which is Lipschitz with respect to $a$.
math.LO↗