Fra\"iss\'e limit via forcing

TL;DR

This paper introduces a forcing method to construct structures of any infinite size from a Fra"issé class, extending the classical Fra"issé construction to uncountable cardinals.

Contribution

It develops a forcing framework that generalizes Fra"issé limits to uncountable structures, broadening the scope of Fra"issé theory.

Findings

Defines a forcing notion for constructing structures of size κ

Extends Fra"issé construction from countable to uncountable structures

Provides a new method for building large homogeneous structures

Abstract

Given a Fra\"{i}ss\'{e} class $\mathcal{K}$ and an infinite cardinal $\kappa,$ we define a forcing notion which adds a structure of size $\kappa$ using elements of $\mathcal{K}$, which extends the Fra\"{i}ss\'{e} construction in the case $\kappa=\omega.$