Erdős Problem #1168

Erdős Problem 1168 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Establish in ZFC, without adding the generalized continuum hypothesis, the negative partition relation \[ \aleph_{\omega+1}\nrightarrow(\aleph_{\omega+1},3,3,\ldots)_{\aleph_0}^{2}. \] Equivalently, seek a countable coloring of pairs from a set of size $\aleph_{\omega+1}$ with no color-zero homogeneous set of that size and no monochromatic triangle in any other color.

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

The desired coloring has an equivalent graph-decomposition formulation. Its positive color classes form countably many triangle-free graphs $G_1,G_2,\ldots$; their union $G$ must have no independent set of size $\kappa=\aleph_{\omega+1}$. Conversely any such sequence supplies a coloring: color an edge by the first graph containing it and color all remaining pairs zero. Taking subgraphs preserves triangle-freeness, so overlaps cause no difficulty.

This formulation rules out one simple strategy under the additional inequality $2^{\aleph_0}<\kappa$. Suppose each $G_i$ were countably colorable, with proper coloring $c_i:V\to\mathbb N$. The vector $(c_i(v))_{i\ge1}$ is a proper coloring of the union with at most $2^{\aleph_0}$ colors. Because $\kappa$ is regular and this number is smaller than $\kappa$, one vector class has size $\kappa$; it is independent in $G$, a contradiction.

Thus, in this cardinal regime, assembling only countably colorable (in particular bipartite) positive classes cannot work. The argument is conditional on the continuum inequality and does not assume it as a theorem of ZFC. It makes no claim that every triangle-free graph is countably colorable.

VERIFICATION AND REMAINING GAP

Construct the required countable decomposition without extra set-theoretic hypotheses. The reformulation and obstruction identify a failed route but do not produce the large-cardinal coloring.

SOURCES

https://www.erdosproblems.com/1168

https://www.erdosproblems.com/history/1168

FINAL

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

COMPLETION ESTIMATE

$4\%$. 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.