Erdős Problem #1170

Erdős Problem 1170 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Is it consistent with ZFC that the ordinal partition relations \[ \omega_2\longrightarrow(\alpha)^2_2 \] hold simultaneously for every ordinal $\alpha<\omega_2$? Thus each two-coloring of pairs from $\omega_2$ would have a monochromatic subset of order type $\alpha$, for each such $\alpha$.

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

ATTACK PLAN AND WORK

ZFC alone gives the restricted relations \[\omega_2\longrightarrow(\alpha)^2_2\qquad(\alpha\le\omega).\tag{1}\] Restrict any coloring to the initial copy of $\omega$. The nested-tail proof of infinite Ramsey's theorem selects an infinite homogeneous subsequence: at each step retain an infinite tail with a constant color to the chosen point, then retain infinitely many chosen points sharing the same associated color. This homogeneous set has order type $\omega$, and taking initial segments proves $1$.

There is also an exact logical reduction of the simultaneous target. For a cofinal set $C\subseteq\omega_2$, having the relation for every $\xi\in C$ is equivalent to having it for every $\alpha<\omega_2$. In one direction this is immediate. In the other, choose $\xi\in C$ above $\alpha$ and take an initial segment of order type $\alpha$ from a homogeneous copy of $\xi$.

This does not reduce the requirement to countably many checks: $\omega_2$ has uncountable cofinality. Nor does an arbitrary homogeneous set of cardinality $\aleph_1$ automatically have any prescribed order type of that cardinality.

VERIFICATION AND REMAINING GAP

Supply a consistency argument giving the relations for a cofinal set of targets below $\omega_2$, or prove an obstruction. Countable homogeneous sequences and monotonicity do not address those higher ordinal targets.

SOURCES

https://www.erdosproblems.com/1170

https://www.erdosproblems.com/latex/1170

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$3\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.