arXiv subjects

Andrea Volpi

Publications and source records attributed to Andrea Volpi.

4 recordsLinked to original sources

arXiv2026

The strength of Ramsey's theorem for $\alpha$-large sets

We calibrate the reverse mathematical strength of a family of extensions of Ramsey's theorem to finite colorings of certain subsets of the natural numbers of unbounded finite dimension. Specifically, we analyze the principles $\mathsf{RT}^{!\alpha}_k$ asserting that every $k$-coloring of the exactly $\alpha$-large subsets of an infinite $X \subseteq \mathbb{N}$ admits an infinite homogeneous set, where $\alpha$-largeness is defined via systems of fundamental sequences in the style of Ketonen and Solovay. For each countable ordinal $\alpha < \Gamma_0$ and each $k \geq 2$, we prove over $\mathsf{RCA}_0$ that the hierarchy of theorems $\mathsf{RT}^{!\a}_k$ corresponds exactly to the hierarchy of systems axiomatized by closure under transfinite Turing jumps, yielding a fine-grained classification between $\mathsf{ACA}_0$ and $\mathsf{ATR}_0$. Our results extend previous work on the case $\alpha=\omega$ and provide a uniform correspondence between countable indecomposable ordinals below $\Gamma_0$ and natural Ramsey-like theorems.

math.LO↗

arXiv2026

Reverse mathematics and dimension of posets

Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this paper we study classical bounding results for order dimension within the framework of reverse mathematics. We focus on principles asserting that the dimension of a poset can be bounded in terms of the dimension of subposets obtained by removing chains or points, denoted by $\mathsf{DBi_n}$, $\mathsf{DBc_n}$, and $\mathsf{DB_p}$. We prove that, over $\mathsf{RCA}_0$, both $\mathsf{DBi_n}$ and $\mathsf{DBc_n}$ are equivalent to $\mathsf{WKL}_0$. To analyze $\mathsf{DB_p}$, we introduce a natural strengthening $\mathsf{DB^+_p}$ and show that both $\mathsf{DB_p}$ and $\mathsf{DB^+_p}$ are provable from $\mathsf{WKL}_0$ and from $\mathsf{I}\Sigma^0_2$, while $\mathsf{B}\Sigma^0_2$ does not suffice to prove $\mathsf{DB^+_p}$. The latter result is obtained by showing that the statement \lq\lq $\mathsf{DB^+_p}$ is computably true\rq\rq\ is equivalent to $\mathsf{I}\Sigma^0_2$.

math.LO↗

arXiv2025

The barrier Ramsey theorem

In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of $\alpha$-largeness, where $\alpha$ is a countable ordinal equipped with a system of fundamental sequences. To extend this approach the more appropriate notion is barrier largeness. Since the complexity of barriers can be measured by countable ordinals, we define Ramsey ordinals and, using appropriate iterations of the Veblen functions, we are able to compute them.

math.LO↗

arXiv2024

Effectiveness and strong graph indivisibility

A relational structure is \emph{strongly indivisible} if for every partition $M = X_0 \sqcup X_1$, the induced substructure on $X_0$ or $X_1$ is isomorphic to $\mathcal{M}$. Cameron (1997) showed that a graph is strongly indivisible if and only if it is the complete graph, the completely disconnected graph, or the random graph. We analyze the strength of Cameron's theorem using tools from computability theory and reverse mathematics. We show that Cameron's theorem is is effective up to computable presentation, and give a partial result towards showing that the full theorem holds in the $\omega$-model $\mathsf{REC}$. We also establish that Cameron's original proof makes essential use of the stronger induction scheme $\mathsf{I}\Sigma^0_2$.

math.LO↗