A short proof of a partion relation for triples (Q1977370)

A very nice elementary proof is given for the following theorem due (essentially) to Erdős and Rado: if \(\varphi\) is an order type with \(\varphi\to(\eta)^1_\omega\) then \(\varphi\to(\omega+n,4)^3\) holds for \(n<\omega\). It is also shown that if \(\varphi\) is an order type and \(\kappa\) an infinite cardinal then \(\varphi\not\to(\omega)^1_{2^\kappa}\) implies \(\varphi\to(\kappa+2,\omega)^3\) and \(\varphi\not\to\left(\text{cf}(\kappa)\right)^1_{\kappa}\) implies \(\varphi\to(\kappa+1,4)^3\).