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 $a$.Comment: Minor revision

Borel circle squaring

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 $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ 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 $\mathbb{Z}^d$.Comment: Minor typos correcte

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction

Diversity of gut microflora is required for the generation of B cell with regulatory properties in a skin graft model

B cells have been reported to promote graft rejection through alloantibody production. However, there is growing evidence that B cells can contribute to the maintenance of tolerance. Here, we used a mouse model of MHC-class I mismatched skin transplantation to investigate the contribution of B cells to graft survival. We demonstrate that adoptive transfer of B cells prolongs skin graft survival but only when the B cells were isolated from mice housed in low sterility "conventional" (CV) facilities and not from mice housed in pathogen free facilities (SPF). However, prolongation of skin graft survival was lost when B cells were isolated from IL-10 deficient mice housed in CV facilities. The suppressive function of B cells isolated from mice housed in CV facilities correlated with an anti-inflammatory environment and with the presence of a different gut microflora compared to mice maintained in SPF facilities. Treatment of mice in the CV facility with antibiotics abrogated the regulatory capacity of B cells. Finally, we identified transitional B cells isolated from CV facilities as possessing the regulatory function. These findings demonstrate that B cells, and in particular transitional B cells, can promote prolongation of graft survival, a function dependent on licensing by gut microflora

Clinical effectiveness of the Manchester Glaucoma Enhanced Referral Scheme

BACKGROUND: Glaucoma referral filtering schemes have operated in the UK for many years. However, there is a paucity of data on the false-negative (FN) rate. This study evaluated the clinical effectiveness of the Manchester Glaucoma Enhanced Referral Scheme (GERS), estimating both the false-positive (FP) and FN rates. METHOD: Outcome data were collected for patients newly referred through GERS and assessed in 'usual-care' clinics to determine the FP rate (referred patients subsequently discharged at their first visit). For the FN rate, glaucoma suspects deemed not requiring referral following GERS assessment were invited to attend for a 'reference standard' examination including all elements of assessment recommended by National Institute for Health and Care Excellence (NICE) by a glaucoma specialist optometrist. A separate 33 cases comprising randomly selected referred and non-referred cases were reviewed independently by two glaucoma specialist consultant ophthalmologists to validate the reference standard assessment. RESULTS: 1404 patients were evaluated in GERS during the study period; 651 (46.3%) were referred to the Hospital Eye Service (HES) and 753 (53.6%) were discharged. The FP rate in 307 assessable patients referred to the HES was 15.5%. This study reviewed 131 (17.4%) of those patients not referred to the HES through the GERS scheme; 117 (89.3%) were confirmed as not requiring hospital follow-up; 14 (10.7%) required follow-up, including 5 (3.8%) offered treatment. Only one patient (0.8%) in this sample met the GERS referral criteria and was not referred (true FN). There were no cases of missed glaucoma or non-glaucomatous pathology identified within our sample. CONCLUSION: The Manchester GERS is an effective glaucoma filtering scheme with a low FP and FN rate