"A partition calculus in set theory" by Erdös and Rado for readers from the twenty-first century
David Joël de Graaf
Abstract:
The seminal paper “A partition calculus in set theory” by Paul Erdös and Richard Rado is notoriously hard to read and contains many interesting results hidden behind outdated notation. This thesis therefore aims at a modernisation of the contents of the paper, in order to make the paper more accessible. This entails rewriting theorems and their proofs using modern mathematical notation. Of particular interest is a key result from the paper, the Positive Stepping Up Lemma, and we conjecture that one instance of the lemma, the partition relation ℶ^+_n → (ω +n+1)^r_m, cannot be improved.
We also adjust the proof of the Negative Stepping Up Lemma in order to prove the implication κ↛(ω^α)^r_m ⇒ 2^κ↛(ω^α)^{r+1}_ m , where κ is an infinite cardinal, α is an ordinal and r, m < ω. We deduce the negative partition relations ℶ^+_n ↛ (ω^2)^{n+3}_2 for all n < ω, providing a bound to the conjecture.