Diamond Open Access Journals at Cambridge

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Recent Articles

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PublicationOpen Access
Dependent measures in independent theories
(2026) Khanaki, Karim
We introduce the notion of dependence, as a property of a Keisler measure, and generalize some results of Hrushovski, Pillay & Simon (2013) in NIP theories (theories satisfying the negation of the independence property) to arbitrary theories. Among other things, we show that this notion is very natural and fundamental for several reasons:
  1. all measures in NIP theories are dependent,
  2. all types and all frequency interpretation measures (fims) in any theory are dependent, and
  3. as a crucial result in measure theory, the Glivenko–Cantelli class of functions (formulas) is characterized by dependent measures.
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PublicationOpen Access
Models of bounded arithmetic and variants of the pigeonhole principle
(2026) Narusevych, Mykyta
We give an elementary proof that theory $T^1_2(R)$ augmented by the weak pigeonhole principle for all $\Delta^{\mathrm{b}}_1(R)$-definable relations does not prove the bijective pigeonhole principle for $R$. This can be derived from known more general results but our proof yields a model of $T^1_2(R)$ in which $\mathsf{ontoPHP}^{n+1}_n(R)$ fails for some nonstandard element $n$ while $\mathsf{PHP}^{m+1}_m$ holds for all $\Delta^{\mathrm{b}}_1(R)$-definable relations and all $m \leq n^{1-\varepsilon}$, where $\varepsilon > 0$ is a fixed standard rational parameter. This can be seen as a step towards solving an open question posed by Ajtai (1990).
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