Extension property for partial automorphisms of the n-partite and semigeneric tournaments

2024-01-22

arXiv.org

AI Summary

This paper investigates the extension property for partial automorphisms (EPPA) and dynamical properties of the universal limits of finite *n*-partite tournaments (*n* ≥ 2) and semi-generic tournaments. Employing methods from model theory—including Fraïssé theory, expansions of locally finite classes, the Ample Generics criterion, and combinatorial analysis—the authors establish two main results: (1) The class of *n*-partite tournaments for all *n* ∈ {2, 3, …, ω} and the class of semi-generic tournaments both satisfy EPPA; (2) Their respective Fraïssé limits possess ample generics, implying that their automorphism groups are rich in structure and topologically abundant. This work not only fully confirms EPPA for two fundamental classes of directed graphs but also provides the first proof of ample generics for both the ω-partite and semi-generic universal tournaments—thereby furnishing a crucial structural foundation for studying minimal flows and automorphism group actions in topological dynamics.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite $n$-partite tournaments for $n in {2,3,ldots,omega}$, and for the class of finite semigeneric tournaments. We also prove that the generic $omega$-partite tournament and the generic semigeneric tournament have ample generics.

Problem

• Proving EPPA for n-partite tournaments
• Establishing EPPA for semigeneric tournaments
• Demonstrating ample generics for generic tournaments

Innovation

• EPPA proof for n-partite tournaments
• EPPA for finite semigeneric tournaments
• Ample generics in generic tournaments