arXiv subjects

Akito Tsuboi

Publications and source records attributed to Akito Tsuboi.

4 recordsLinked to original sources

arXiv2026

On chromatic number of countable graphs

This paper investigates when countable graphs have a finite or an infinite chromatic number through model theoretic methods. For Fra\"{i}ss\'{e} limits, we show that instability forces the chromatic number to be infinite, yielding a complete classification of homogeneous graphs with a finite chromatic number. In contrast, Hrushovski construction always produces graphs of finite chromatic number, though the value can be made arbitrarily large. In tame settings -- such as stable graphs of $U$-rank one and graphs definable in o-minimal structures -- an infinite chromatic number necessarily yields arbitrarily large cliques. These results provide a unified framework connecting structural model theoretic properties with chromatic behavior.

math.LO↗

arXiv2026

Elementary extensions of almost o-minimal structures

This paper investigates almost o-minimal structures, a weakening of o-minimality introduced by Fujita to capture structures that lie outside the classical o-minimal framework. In contrast to o-minimality and local o-minimality, almost o-minimality is not preserved under elementary equivalence. This raises the natural question of whether every almost o-minimal structure admits a proper elementary extension that is again almost o-minimal. The main result of this paper provides an affirmative answer to this question.

math.LO↗

arXiv2020

On the number of independent orders

We investigate a model theoretic invariant $κ_{srd}^m(T)$, which was introduced by Shelah in his famous book, and prove that $κ_{srd}^m(T)$ is sub-additive. When $κ_{srd}^m(T)$ is infinite, this gives the equality $κ^m_{srd}(T)=κ^1_{srd}(T)$, answering a question by Shelah. We apply the same proof method to analyze another invariant $κ^m_{ird}(T)$, and show that it is also sub-additive, improving a result in the book.

math.LO↗

arXiv2001

Definability of initial segments

We consider implicit definability of the standard part {0,1,...} in nonstandard models of Peano arithmetic (PA), and we ask whether there is a model of PA in which the standard part is implicitly definable. In section 1, we define a certain class of formulas, and show that in any model of PA the standard part is not implicitly defined by using such formulas. In section 2 we construct a model of PA in which the standard part is implicitly defined. To construct such a model, first we assume a set theoretic hypothesis diamondsuit_{S_lambda^{lambda^+}}, which is an assertion of the existence of a very general set. Then we shall eliminate the hypothesis using absoluteness for the existence of a model having a tree structure with a certain property.

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