A strongly non-Ramsey order type (Q1271912)

Given two order types \(\psi\), \(\theta\) and a cardinal \(\kappa\), \(\psi \nrightarrow[\theta]^2_{\kappa}\) asserts the existence of an ordered set \((X,<)\) of order type \(\psi\) and a function \(f:[X^2]\to\kappa\) such that for every subset \(Y\) of \(X\) of order type \(\theta\), the range of \(f\) on \([Y]^2\) is all of \(\kappa\). It is shown that if a Cohen real is added to a model of the Souslin Hypothesis, then in the extension there is an order type \(\theta\) of cardinality \(\aleph_1\) such that \(\psi \nrightarrow [\theta]^2_{\omega}\) for every order type \(\psi\). Moreover, it is consistent that there is an order type \(\theta\) of cardinality \(\aleph_1\) such that \(\psi\nrightarrow [\theta]^2_{\omega_1}\) for every order type \(\psi\). The authors observe that the following statement was shown consistent by \textit{S. Shelah} [Lect. Notes Math. 1401, 167-193 (1989; Zbl 0683.04002)]: Given an order type \(\theta\) and a cardinal \(\kappa\), there is an order type \(\psi\) with the following property: for every ordered set \((X, <)\) of order type \(\psi\) and every \(f:[X]^2\to \kappa\), there is a subset \(Y\) of \(X\) of order type \(\theta\) such that the range of \(f\) on \([Y]^2\) has size at most 2.