We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.

We provide a new criterion for embedding.

We show that there is an antichain basis for neither the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.

I argue that we cannot fully understand Aristotle’s position on political stability and state preservation in the Politics with paying close attention to his Eudemian Ethics. We learn from considering the Politics and the Eudemian Ethics in concert that even ‘correct’ regimes are unstable when citizens do not possess full virtue. Aristotle introduces his formal account of the knowledge requirements for virtue in Eudemian Ethics 8.3, and he applies these knowledge requirements as an explanation for state decline in Politics 2.9 (...) when discussing the Spartans. If we primarily focus on the Nicomachean Ethics as Aristotle’s single essential ethical work, we will not learn the lesson he intends his readers to take away from the Spartan discussion in the Politics: that virtue requires correct understanding of the hierarchy and structure of the good life. This knowledge prevents the erosion of the virtues of character and the decline of political regimes. (shrink)

We provide a finite basis for the class of Borel functions that are not in the first Baire class, as well as the class of Borel functions that are not $\sigma $ -continuous with closed witnesses.

We generalize Kada’s definable strengthening of Dilworth’s characterization of the class of quasi-orders admitting an antichain of a given finite cardinality.

BackgroundSarcomas are a diverse group of neoplasms that vary greatly in clinical presentation and responsiveness to treatment. Given the differences in the sites of involvement, rarity, and treatment modality, a multidisciplinary approach is required. Previous literature suggests patients with sarcoma suffer from poorer quality of life especially physical and functional wellbeing. Adolescent and young adult patients are an underrepresented population in cancer research and have differing factors influencing QoL.MethodsRetrospective analysis of Young Adult patients enrolled in the Sarcoma Tissue Repository at (...) University of Iowa. QoL was assessed using the self-report FACT-G questionnaire at enrollment and 12 months post-diagnosis; overall scores and the 4 wellbeing subscales were calculated. Linear mixed effects models were used to measure the association between the rate of change in FACT-G subscale scores and baseline clinical, comorbidity, and treatment characteristics.Results49 patients were identified. 57.1% of patients had a malignancy involving an extremity. Mean FACT-G scores of overall wellbeing improved from baseline to 12 months. Social and emotional wellbeing did not differ significantly between baseline and 12 months. Physical wellbeing and functional wellbeing scores improved from baseline to 12 months. No difference was seen for FACT-G overall scores for age, sex, laterality, marital status, performance status, having children, clinical stage, limb surgery, chemotherapy, or tumor size. A difference was demonstrated in physical wellbeing scores for patients with baseline limitation compared to those with no baseline limitation. A difference was demonstrated in social wellbeing based on anatomical site.ConclusionYoung adults with sarcoma treated at a tertiary center had improvements in overall reported QoL at 12 months from diagnosis. Overall baseline QoL scores on FACT-G were lower than the general adult population for YA patients with sarcoma but at 12 months became in line with general population norms. The improvements seen merit further investigation to evaluate how these change over the continuum of care. Quality of life changes may be useful outcomes of interest in sarcoma trials. (shrink)

We show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special (...) kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation. (shrink)

We give classical proofs, strengthenings, and generalizations of Lecomte’s characterizations of analytic ω-dimensional hypergraphs with countable Borel chromatic number.

We establish Hjorth's theorem that there is a family of continuum-many pairwise strongly incomparable free actions of free groups, and therefore a family of continuum-many pairwise incomparable treeable equivalence relations.

We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [N]N, although its Borel chromatic number is N₀. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In (...) particular, this holds for "the" minimum analytic graph G₀ with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ ∈ {2, 3...., N₀. c}, there is a treeing of E₀ with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm—Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (R<N, ⊇). (shrink)

We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.