Reused writeup on Math-983 (gpt pro 5.2)

Erdős Problem 919 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/919

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/919.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/919.tex.

1) FORMAL RESTATEMENT

Let $\omega_2$ denote the first ordinal of cardinality $\aleph_2$. Let $\omega_2^2$ denote the ordinal product $\omega_2\cdot\omega_2$, which is order-isomorphic to $\omega_2\times\omega_2$ with the lexicographic order. A subset X of this well-ordered set has “lesser type” than $\omega_2^2$ if its induced order type is an ordinal <$\omega_2^2$.

Ambiguity: the problem statement speaks of “subgraph whose vertices have a lesser type”. We interpret this as: for every subset X$\subseteq$ $\omega_2^2$ with order type <$\omega_2^2$, the induced subgraph G[X] has $\chi(G[X])\le\aleph_0$.

Question (A): Does there exist a graph G with vertex set $\omega_2^2$ and $\chi(G)$=$\aleph_2$ such that for every X$\subseteq$ $\omega_2^2$ with order type <$\omega_2^2$ we have $\chi(G[X])\le$ $\aleph_0$?

Question (B): Same but requiring $\chi(G)$=$\aleph_1$.

2) QUICK LITERATURE/CONTEXT CHECK

The problem statement records: - Babai: if G is a graph on a well-ordered vertex set with $\chi(G)\ge$ $\aleph_0$ then there is a subgraph on vertices of order type $\omega$ with chromatic number $\aleph_0$. - Erdős–Hajnal constructed a graph on $\omega_1^2$ with chromatic number $\aleph_1$ such that every strictly smaller subgraph has chromatic number $\le\aleph_0$, using the product-order comparability graph: vertices $\alpha,\beta$ and edges when ($\alpha_1$<$\alpha_2$ and $\beta_1$<$\beta_2$) (symmetrized). - A similar construction yields a graph on $\omega_2^2$ with chromatic number $\aleph_2$ such that every smaller subgraph has chromatic number $\le$ $\aleph_1$.

I do not use or assert any additional external results beyond what is stated in the problem file.

3) ATTACK PLAN

Proof-track (existence): - Try to strengthen the Erdős–Hajnal product-order construction to force countable chromatic number on all vertex sets of order type <$\omega_2^2$ (not just $\le\aleph_1$). - Attempt to build G by recursion along the well-order, ensuring each initial segment (hence each smaller-type set) stays countably colorable, while the whole graph becomes $\aleph_2$-chromatic.

Disproof-track (nonexistence): - Attempt to prove that if $\chi(G)$=$\aleph_2$ on $\omega_2^2$, then some smaller-type subset must already witness uncountable chromatic number, perhaps via a reflection/pressing-down argument on the lex order.

Best path here: I cannot complete either track. I therefore prove concrete properties of the Erdős–Hajnal comparability graph (since it is the natural candidate) and isolate the gap to reach $\le\aleph_0$ on all smaller types.

4) WORK

Definition (Erdős–Hajnal product-order graph). Fix an infinite cardinal $\kappa$ and consider vertex set V=$\kappa\times\kappa$. Define an undirected graph $H_\kappa$ on V by putting an edge between distinct vertices $\alpha_1,\beta_1$ and $\alpha_2,\beta_2$ iff either ($\alpha_1$<$\alpha_2$ and $\beta_1$<$\beta_2$) or ($\alpha_2$<$\alpha_1$ and $\beta_2$<$\beta_1$). Equivalently, there is an edge iff the two vertices are comparable in the strict product order on $\kappa\times\kappa$.

Lemma 919.1 (chromatic number of $H_\kappa$ equals $\kappa$). For every infinite cardinal $\kappa$, $\chi$$H_\kappa$=$\kappa$.

Proof. Upper bound: Color each vertex $\alpha,\beta$ by its first coordinate $\alpha$. If two vertices share the same first coordinate $\alpha$, then neither can have $\alpha_1$<$\alpha_2$, so they are not adjacent. Hence each color class is independent and this is a proper coloring using $\kappa$ colors. Thus $\chi$$H_\kappa$$\le$ $\kappa$.

Lower bound: Consider the diagonal set D:={$\alpha,\alpha$:$\alpha$<$\kappa$}. If $\alpha,\alpha$ and $\alpha',\alpha'$ are distinct, then either $\alpha$<$\alpha$' and also $\alpha$<$\alpha$', hence they are adjacent. So D induces a clique of size $\kappa$. Any proper coloring of a clique of size $\kappa$ uses at least $\kappa$ colors, hence $\chi$$H_\kappa$$\ge$ $\kappa$. Combining bounds gives $\chi$$H_\kappa$=$\kappa$. $\square$

Lemma 919.2 (countable-coordinate-bounded subgraphs are countably colorable). Let $\kappa$ be any cardinal and let S$\subseteq$ $\kappa\times\kappa$. If there exists a countable ordinal $\gamma$<$\omega_1$ such that S$\subseteq$ $\gamma\times\kappa$, then $\chi$$H_\kappa[S]$$\le$ $\aleph_0$. Similarly, if S$\subseteq$ $\kappa\times\gamma$ for some countable $\gamma$, then $\chi$$H_\kappa[S]$$\le$ $\aleph_0$.

Proof. Assume S$\subseteq$ $\gamma\times\kappa$ with $\gamma$ countable. Define c:S$\to\gamma$ by c$\alpha,\beta$=$\alpha$. As in Lemma 919.1, vertices with the same first coordinate are non-adjacent, so each fiber c^{-1}$\alpha$ is independent. Since $\gamma$ is countable, this gives a proper coloring using at most $\aleph_0$ colors. The second statement (bounding the second coordinate) is proved by coloring with the second coordinate. $\square$

FAST REALITY CHECK. A finite analogue of $H_\kappa$ on [m]$\times$[m] (m finite) has chromatic number exactly m: - The diagonal gives a clique of size m, so $\chi\ge$ m. - Coloring by the first coordinate uses m colors, so $\chi\le$ m. This matches Lemma 919.1 in the finite case.

5) VERIFICATION

- Lemma 919.1: Verified that “color by first coordinate” is valid because adjacency requires strict inequality in the first coordinate. - Lemma 919.2: Verified that the number of colors used is |$\gamma$|$\le\aleph_0$ and each color class is independent. - The lemmas do not address the key “lesser type” condition; they only show countable chromaticity for a restricted family of smaller-type sets (those bounded by a countable ordinal in one coordinate).

6) FINAL

UNRESOLVED (i) Strongest proved partial result: The natural candidate graph $H_{\omega_2}$ (product-order comparability graph on $\omega_2\times\omega_2$) has $\chi$$H_{\omega_2}$=$\aleph_2$ (Lemma 919.1). Moreover, any induced subgraph whose vertices lie in $\gamma\times\omega_2$ or $\omega_2\times\gamma$ for some countable $\gamma$ is countably colorable (Lemma 919.2). (ii) First gap (crisp): Show (or refute) that there exists a graph on $\omega_2^2$ with chromatic number $\aleph_2$ for which every subset of vertices of order type <$\omega_2^2$ induces a countably colorable subgraph. (iii) Top 3 next moves: 1. Analyze the Erdős–Hajnal construction more deeply: characterize which order types <$\omega_2^2$ already force uncountable chromatic number in $H_{\omega_2}$. This becomes a concrete test case for (A). 2. Attempt to build a graph by recursion along the lex order on $\omega_2^2$, explicitly maintaining countable colorings on all initial segments while diagonalizing against all $\aleph_0$-colorings globally. 3. Try to prove a reflection lemma: if $\chi(G)$=$\aleph_2$ on $\omega_2^2$ then there exists X of order type <$\omega_2^2$ with $\chi(G[X])\ge\aleph_1$. Identifying the first nontrivial case (e.g. order type $\omega_2\cdot\omega$) would already be progress. (iv) Minimal counterexample structure: A positive example for (A) would be a graph that is globally $\aleph_2$-chromatic on the well-order $\omega_2^2$ but whose chromatic number “collapses” to countable on every proper initial-type subset. Any disproof must show that some smaller-type subset necessarily inherits uncountable chromaticity.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.