Erdős Problem #1218

Erdős Problem 1218 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Assume the Generalised Continuum Hypothesis. Put $\beta=\omega_{\omega+1}$, with $\omega_\alpha$ denoting the initial ordinal of cardinality $\aleph_\alpha$. Does the negative partition relation \[\aleph_{\beta+1}\not\longrightarrow(\aleph_\beta,4,4,\ldots)^3\] hold, where the list has countably infinitely many entries equal to $4$? Explicitly, the question asks for a colouring of the unordered triples from a set of size $\aleph_{\beta+1}$ by colours $0,1,2,\ldots$, with no colour-$0$ homogeneous subset of size $\aleph_\beta$ and no colour-$j$ homogeneous four-element subset for any $j\geq1$.

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

Put $\kappa=\aleph_\beta$ and $\theta=\kappa^+$. The desired coloring is equivalent to a countable family of $K_4^{(3)}$-free 3-uniform hypergraphs on $\theta$ vertices whose union has no independent set of size $\kappa$. The positive color classes give such a family. Conversely color each covered triple by its first covering hypergraph, and all other triples zero; each positive class remains $K_4^{(3)}$-free as a subhypergraph.

Under the stated GCH, this construction cannot use only countably colorable positive hypergraphs. If each has a proper weak coloring $c_j:V\to\mathbb N$ (no monochromatic edge), the vector $(c_j(v))_{j\ge1}$ colors the union with at most $2^{\aleph_0}=\aleph_1$ colors. Regularity of $\theta>\aleph_1$ implies one vector class has size $\theta$ and contains no edge of the union, contradicting the required absence of even a $\kappa$-sized independent set.

The distinction between being $K_4^{(3)}$-free and being countably colorable is therefore essential. Substituting the latter property merely because it is easy to arrange would impose an obstruction incompatible with the target. The proof uses GCH only for the small size of the vector-color range.

VERIFICATION AND REMAINING GAP

Construct the countable covering family with sufficiently high chromatic complexity, or otherwise establish the negative partition relation. The reformulation and obstruction do not supply that family.

SOURCES

https://www.erdosproblems.com/1218

https://www.erdosproblems.com/latex/1218

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.