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Math-864-A1 · revision r1 · on Math-864

Reused writeup on Math-864 (gpt pro 5.2)

No claim recorded · LLM-generated · gpt pro 5.2 ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem 593 --- collection record
  2. PROVENANCE AND STATUS
  3. REUSED SOURCE 1
  4. COMPLETION ESTIMATE

Erdős Problem 593 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/593

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_5.2/593.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/593.tex.

Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $>\aleph_0$. Similar problems were investigated by Erdős, Galvin, and Hajnal [EGH75]. Erdős claims that for graphs the problem is completely solved: a graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle. References [EGH75] Erdős, P. and Galvin, F. and Hajnal, A., On set-systems having large chromatic number and not containing prescribed subsystems . $1975$, 425--513.

FORMAL RESTATEMENT

A  3-uniform hypergraph is a pair $H=(V,E)$ where $V$ is a set of vertices and $E\subseteq [V]^3$ is a set of 3-element subsets (hyperedges). A  proper vertex-colouring of $H$ with a set of colours $\Gamma$ is a map $f:V\to\Gamma$ such that no hyperedge $e\in E$ is monochromatic, i.e. $f$ is not constant on $e$. The  chromatic number $\chi(H)$ is the least cardinal $\kappa$ for which there is a proper colouring with $\kappa$ colours.

A finite 3-uniform hypergraph $F$ appears in $H$ if there is an injective map $\varphi:V(F)\to V(H)$ such that for every edge $\{a,b,c\}\in E(F)$, the image $\{\varphi(a),\varphi(b),\varphi(c)\}$ lies in $E(H)$ (i.e. $F$ is isomorphic to a subhypergraph of $H$).

Problem: characterize all finite 3-uniform hypergraphs $F$ such that for every 3-uniform hypergraph $H$ with $\chi(H)>\aleph_0$, $F$ appears in $H$.

QUICK LITERATURE/CONTEXT CHECK

I will not import external results beyond what is explicitly stated in the problem text. The text gives an analogy for graphs: uncountably chromatic graphs contain every finite bipartite graph but can avoid any fixed odd cycle. The hypergraph analogue is posed as an open-ended characterization problem.

ATTACK PLAN

Proof-track ideas (unavoidability).

  • Prove general "compactness" consequences of $\chi(H)>\aleph_0$, e.g. that $H$ contains finite subhypergraphs of arbitrarily large finite chromatic number, and then try to infer that certain finite configurations must embed.
  • Identify a candidate class of "unavoidable" finite 3-graphs (hypergraph analogue of bipartite graphs), perhaps those with a certain 2-colourability/acyclicity property, and try to embed them via a recursive construction.

Disproof-track ideas (avoidability).

  • For a given finite $F$, attempt to construct a 3-uniform hypergraph $H$ with $\chi(H)>\aleph_0$ but $F$-free, by imposing intersection restrictions (e.g. linearity) or large-girth type constraints while keeping uncountable chromatic number.

I did not reach a characterization. I record two rigorous lemmas forced by the hypothesis $\chi(H)>\aleph_0$.

WORK

FAST REALITY CHECK

  • If $\chi(H)>\aleph_0$, then in particular $\chi(H)>2$ and $\chi(H)>k$ for every finite $k$. Thus $H$ cannot be "matching-like" (disjoint edges), and must contain finite subconfigurations that force more than $k$ colours.

Lemma 1 (finite-$k$ compactness for hypergraph colourings). Let $H=(V,E)$ be a (not necessarily finite) hypergraph and let $k\in\mathbb{N}$. If every finite subhypergraph of $H$ is $k$-colourable, then $H$ is $k$-colourable.

Proof. Let $[k]:=\{1,2,\dots,k\}$ and consider the product space $X:=[k]^V$ of all functions $f:V\to[k]$, equipped with the product topology where $[k]$ has the discrete topology. Since $[k]$ is finite and discrete, it is compact, and therefore by Tychonoff's theorem $X$ is compact.

For each hyperedge $e=\{v_1,\dots,v_r\}\in E$ (here $r=3$ in our application), define \[ C_e:=\{f\in X: \text{$f$ is \emph{not} constant on $e$}\}. \] Then $C_e$ is open (indeed clopen) in $X$: it is the complement of the closed set where $f(v_1)=\cdots=f(v_r)$. A function $f\in X$ is a proper $k$-colouring of $H$ exactly when $f\in\bigcap_{e\in E} C_e$.

We claim the family $\{C_e:e\in E\}$ has the finite intersection property. Indeed, take finitely many edges $e_1,\dots,e_m$. Let $V_0\subseteq V$ be the finite set of vertices appearing in these edges. By assumption, the finite subhypergraph induced on $V_0$ is $k$-colourable, so there exists $g:V_0\to[k]$ which is not constant on any $e_i$. Extend $g$ arbitrarily to a function $f:V\to[k]$. Then $f\in C_{e_i}$ for each $i$, so $f\in\bigcap_{i=1}^m C_{e_i}$. Thus every finite intersection is nonempty.

By compactness of $X$, the intersection $\bigcap_{e\in E} C_e$ is nonempty. Any $f$ in this intersection is a proper $k$-colouring of $H$.$\square$

Corollary 1. If a hypergraph $H$ is not $k$-colourable for some finite $k$, then $H$ has a finite subhypergraph which is not $k$-colourable. In particular, if $\chi(H)>\aleph_0$ then for every finite $k$ there exists a finite subhypergraph $F_k\subseteq H$ with $\chi(F_k)>k$.

Proof. The first statement is the contrapositive of Lemma lem:hypergraph_compactness. For the second, if $\chi(H)>\aleph_0$ then $H$ is not $k$-colourable for any finite $k$, so apply the first statement to each $k$.$\square$

Lemma 2 (a forced finite configuration: intersecting edges). Let $H=(V,E)$ be a 3-uniform hypergraph. If the edges of $H$ are pairwise disjoint (i.e. $e\cap e'=\emptyset$ for all distinct $e,e'\in E$), then $\chi(H)\le 2$. Consequently, if $\chi(H)>2$ (and hence if $\chi(H)>\aleph_0$), then $H$ contains two edges with nonempty intersection.

Proof. Assume edges are pairwise disjoint. Define a 2-colouring $f:V\to\{\text{red},\text{blue}\}$ as follows. For each edge $e=\{a,b,c\}\in E$, set $f(a)=\text{red}$ and $f(b)=f(c)=\text{blue}$. This is consistent because distinct edges are disjoint, so no vertex receives two different prescriptions. For vertices not contained in any edge, assign (say) $\text{red}$. Then every edge contains both colours, so no edge is monochromatic. Thus $f$ is a proper 2-colouring and $\chi(H)\le 2$. The final sentence is the contrapositive.$\square$

VERIFICATION

  • Lemma lem:hypergraph_compactness uses only finite-$k$ compactness; it does not extend to $k=\aleph_0$, so it is compatible with the existence of hypergraphs of chromatic number $\aleph_0$ whose finite subhypergraphs are all 2-colourable.
  • Lemma lem:intersecting_edges is checked on the boundary case $E=\emptyset$ (then $\chi(H)=1\le 2$).

FINAL

UNRESOLVED

(i) Strongest proved partial result. If $\chi(H)>\aleph_0$, then for every finite $k$ the hypergraph contains a finite subhypergraph of chromatic number $>k$ (Corollary cor:finite_obstructions); and in particular $H$ must contain two intersecting edges (Lemma lem:intersecting_edges).

(ii) First gap (crisp). Find a necessary-and-sufficient intrinsic property of a finite 3-uniform hypergraph $F$ guaranteeing that every 3-uniform hypergraph $H$ with $\chi(H)>\aleph_0$ contains an isomorphic copy of $F$. Even the candidate criterion "$F$ is 2-colourable" (hypergraph analogue of bipartite) is not proved here.

(iii) Top 3 next moves.

  1. Test the conjectural analogue "unavoidable iff 2-colourable" by attempting: (a) an embedding lemma for every finite 2-colourable 3-graph into any $H$ with $\chi(H)>\aleph_0$, and (b) for each non-2-colourable $F$ construct an $F$-free $H$ with $\chi(H)>\aleph_0$.
  2. Build explicit candidate $H$ with $\chi(H)=\aleph_1$ and strong intersection restrictions (e.g. linear 3-graphs) and check which small $F$ embed.
  3. Perform small-configuration searches inside known high-chromatic finite 3-graphs to guess the "unavoidable" class and then attempt to lift to the uncountable setting.

(iv) Minimal counterexample structure. To disprove a candidate "unavoidable" finite $F$, one would need a 3-uniform hypergraph $H$ with $\chi(H)>\aleph_0$ but $F\not\subseteq H$. Any such $H$ must still contain finite subhypergraphs with arbitrarily large finite chromatic number, so $H$ cannot be too sparse globally; the obstruction must instead come from forbidding a specific local intersection pattern encoded by $F$.

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.