A permutation group acting transitively on certain collections of models

Key Points

• To explore the existence of a dense subgroup in the symmetric group with transitive action on relational structures under Martin's Axiom.
• Utilized σ-centered Martin's Axiom to derive findings.
• Analyzed the natural action of the symmetric group on relational structures.
• Investigated specific models from questions posed by Peter M. Neumann.
• Established the existence of a proper dense subgroup in the symmetric group acting transitively.
• Provided consistent positive answers to unresolved questions from the 1980s concerning relational structures.