# Infinite combinatorics plain and simple

## TL;DR

This paper introduces a simplified method using trees of elementary submodels for infinite combinatorics, extending previous techniques to models of size continuum, and demonstrating broad applications across various combinatorial and topological results.

## Contribution

It develops a new approach based on countably closed models of size continuum, broadening the elementary submodel technique for infinite combinatorics proofs.

## Key findings

- Simplified proofs for paradoxical decompositions of the plane
- Applications to coloring sparse set systems
- Results on graph chromatic number and point-set topology

## Abstract

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.