Reused writeup on Math-982 (gpt pro 5.2)

Erdős Problem 918 --- collection record

PROVENANCE AND STATUS

TeX syntax corrected for display; mathematical content unchanged. Official problem: https://www.erdosproblems.com/918

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 with TeX syntax corrections for display.

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_5.2/918.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/918.tex.

1) FORMAL RESTATEMENT

We treat graphs as simple undirected graphs (no loops, no multiple edges). For a graph G=(V,E), the chromatic number $\chi(G)$ is the least cardinal $\kappa$ such that V can be written as a disjoint union of $\kappa$ independent sets.

Ambiguity: the problem statement says “subgraph on $\aleph_1$ vertices”. For an upper bound condition “$\chi$$\cdot$$\le\aleph_0$”, this is equivalent to requiring the induced subgraph on that vertex set to have $\chi\le\aleph_0$ (since deleting edges cannot increase chromatic number). Accordingly, in what follows “subgraph on X$\subseteq$ V” means the induced subgraph G[X].

Question (A): Does there exist a graph G with |V|=$\aleph_2$ and $\chi(G)$=$\aleph_2$ such that for every X$\subseteq$ V with |X|=$\aleph_1$ we have $\chi(G[X])\le$ $\aleph_0$?

Question (B): Does there exist a graph H with |V(H)|=$\aleph_{\omega+1}$ and $\chi(H)$=$\aleph_1$ such that for every Y$\subseteq$ V(H) with |Y|=$\aleph_{\omega}$ we have $\chi(H[Y])\le$ $\aleph_0$?

Edge cases/conventions: We use $\aleph_0$=|$\mathbb N$|. For induced subgraphs of size $\aleph_1$, “$\chi\le$ $\aleph_0$” means “countably colorable”.

2) QUICK LITERATURE/CONTEXT CHECK

The problem statement itself records: - Erdős–Hajnal proved: for every finite k there exists a graph with chromatic number $\aleph_1$ in which every subgraph on fewer than $\aleph_k$ vertices has chromatic number $\le$ $\aleph_0$. - A variant in [Er69b] with “=$\aleph_0$” for every subgraph is (if interpreted as non-induced subgraph) trivially impossible because one can delete all edges; hence the intended formulation is the “$\le\aleph_0$” version above.

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

3) ATTACK PLAN

Proof-track ideas (existence): - Try to build G via transfinite recursion/forcing so that global chromatic number is $\aleph_2$ but any $\aleph_1$-sized induced subgraph admits a countable coloring (“non-reflection” of chromatic number). - Try to encode a strong anti-chain/club-guessing structure into edges while controlling all $\aleph_1$-substructures.

Disproof-track ideas (nonexistence in ZFC): - Attempt to prove a “reflection” theorem: if $\chi(G)$=$\aleph_2$ on $\aleph_2$ vertices, then some $\aleph_1$-sized induced subgraph must have uncountable chromatic number. - Attempt to derive an $\aleph_1$-sized $K_{\aleph_1}$ (or another uncountably-chromatic configuration) inside any $\aleph_2$-chromatic graph.

Best path here: I do not know a route to a complete proof or counterexample. I therefore record rigorous structural consequences (lemmas) and isolate the precise gap.

4) WORK

Lemma 918.1 (no uncountable clique). Assume G is a graph such that for every X$\subseteq$ V(G) with |X|=$\aleph_1$ we have $\chi(G[X])\le$ $\aleph_0$. Then G contains no clique of size $\aleph_1$.

Proof. Suppose for contradiction that C$\subseteq$ V(G) is a clique and |C|=$\aleph_1$. Then in the induced subgraph G[C], every pair of distinct vertices is adjacent. In any proper coloring of G[C], distinct vertices must receive distinct colors, hence $\chi(G[C])$=|C|=$\aleph_1$. This contradicts the assumption $\chi(G[C])\le$ $\aleph_0$. $\square$

Lemma 918.2 (every $\aleph_1$-set contains an $\aleph_1$ independent set). Assume G is a graph such that for every X$\subseteq$ V(G) with |X|=$\aleph_1$ we have $\chi(G[X])\le$ $\aleph_0$. Then for every X$\subseteq$ V(G) with |X|=$\aleph_1$ there exists an independent set I$\subseteq$ X with |I|=$\aleph_1$.

Proof. Fix X$\subseteq$ V(G) with |X|=$\aleph_1$. By assumption, $\chi(G[X])\le$ $\aleph_0$, so there exists a coloring c:X$\to$ $\mathbb N$ such that each color class c^{-1}(n) is independent. If every color class c^{-1}(n) were countable, then X=$\bigcup_{n\in\mathbb N}$ c^{-1}(n) would be a countable union of countable sets and hence countable. But X has size $\aleph_1$, contradiction. Therefore there exists n$\in\mathbb N$ such that c^{-1}(n) is uncountable. Since any uncountable subset of $\omega_1$ has cardinality $\aleph_1$, we conclude |c^{-1}(n)|=$\aleph_1$. Let I:=c^{-1}(n). Then I$\subseteq$ X is independent and |I|=$\aleph_1$. $\square$

Lemma 918.3 (singular-cardinal variant: large independent sets of some $\aleph_n$). Let $\alpha\in$$0,1$ be irrelevant here; this lemma concerns the second question. Assume H is a graph such that for every Y$\subseteq$ V(H) with |Y|=$\aleph_\omega$ we have $\chi(H[Y])\le$ $\aleph_0$. Then for every such Y and every fixed n$\in\mathbb N$, there exists an independent set I$\subseteq$ Y with |I|$\ge$ $\aleph_n$.

Proof. Fix Y$\subseteq$ V(H) with |Y|=$\aleph_\omega$ and a proper coloring c:Y$\to\mathbb N$. Write Y=$\bigcup_{m\in\mathbb N}$ C_m where C_m:=c^{-1}(m) is independent. Fix n$\in\mathbb N$. If every C_m had cardinality <$\aleph_n$, then since $\aleph_n$ is regular and $\mathbb N$ is countable, we would have \[ |Y|=\Big|\bigcup_{m\in\mathbb N} C_m\Big|<\aleph_n\cdot\aleph_0=\aleph_n, \] contradicting |Y|=$\aleph_\omega\ge$ $\aleph_n$. Hence there exists m with |C_m|$\ge$ $\aleph_n$. Take I:=C_m. $\square$

FAST REALITY CHECK. There is no finite “toy” analogue that captures the set-theoretic content, but two sanity checks are: - If one replaces “$\aleph_1$” by “finite”, de Bruijn–Erdős compactness prevents $\chi(G)$ from being larger than the maximum finite chromatic number of its finite subgraphs. - Here the obstruction is genuinely at the level of uncountable subgraphs, so compactness does not apply.

5) VERIFICATION

- Lemma 918.1: Verified that a clique of size $\aleph_1$ forces chromatic number $\aleph_1$ (each vertex must be a distinct color). - Lemma 918.2: Verified the only set-theoretic input is: a countable union of countable sets is countable. This is standard in ZFC. Also used: any uncountable subset of $\omega_1$ has size $\aleph_1$. - Lemma 918.3: Checked cardinal arithmetic: for regular $\aleph_n$, a countable union of sets each of size <$\aleph_n$ has size <$\aleph_n$, so cannot reach $\aleph_\omega$. - No hidden use of induced vs non-induced: in both lemmas we use induced subgraphs/colorings of the vertex set.

6) FINAL

UNRESOLVED (i) Strongest proved partial result: Under the “every $\aleph_1$-induced-subgraph is countably colorable” hypothesis, every $\aleph_1$-sized vertex set contains an independent subset of size $\aleph_1$ (Lemma 918.2) and in particular there is no clique of size $\aleph_1$ (Lemma 918.1). For the $\aleph_\omega$-sized subgraph hypothesis, one can still force the existence of independent sets of size $\aleph_n$ for every fixed n (Lemma 918.3). (ii) First gap (crisp): Construct in ZFC (or disprove in ZFC) a graph G with |V|=$\aleph_2$ and $\chi(G)$=$\aleph_2$ such that $\chi(G[X])\le\aleph_0$ for every X$\subseteq$ V with |X|=$\aleph_1$. (iii) Top 3 next moves: 1. Try to formalize the property of Lemma 918.2 as a “large independent set reflection” and see whether it forces $\chi(G)\le\aleph_1$ by a transfinite coloring recursion. 2. Attempt a forcing-style construction: build edges along a $\square_{\aleph_1}$-sequence or club-guessing sequence while explicitly maintaining countable colorings on all $\aleph_1$ stages. 3. Attempt a disproof under additional reflection principles (e.g. show that if certain stationary reflection holds then any $\aleph_2$-chromatic graph reflects to an $\aleph_1$-chromatic induced subgraph), and isolate which step uses extra axioms. (iv) Minimal counterexample structure: A counterexample-to-nonexistence would be a graph on $\aleph_2$ vertices of chromatic number $\aleph_2$ that nevertheless satisfies the very strong property “every $\aleph_1$-subset contains an $\aleph_1$ independent set”. Any ZFC disproof would have to show these two requirements are incompatible.

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.