A Statement in Combinatorics that is Independent of ZFC (an exposition)

TL;DR

This paper discusses a statement in combinatorics related to coloring the reals and its independence from ZFC, providing an exposition of Erdős's proof linking it to the negation of the Continuum Hypothesis.

Contribution

It offers an accessible exposition of Erdős's proof that the combinatorial statement is independent of ZFC, connecting it to the continuum hypothesis.

Findings

The statement S is equivalent to the negation of CH.

S is independent of ZFC.

Modern observations relate to results of similar independence.

Abstract

It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which we denote S: For every $\aleph_0$-coloring of the reals there exists distinct reals $e_1,e_2,e_3,e_4$ such that $e_1+e_2=e_3+e_4$?} Is it true? Erdos showed that S is equivalent to the negation of the Continuum Hypothesis, and hence S is indepedent of ZFC. We give an exposition of his proof and some modern observations about results of this sort.