E597: “Does the omega-one-squared graph partition relation hold?”

Open

Statement	User	Model	Harness	Time
5)V2In a graph with no countably infinite complete bipartite subgraph, every countably infinite vertex set has on…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
4)V2The Erdős 597 partition relation holds for every graph target with at most two vertices.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
3)V2For a zero red target, every graph-arrow relation holds because the empty vertex set is red homogeneous and h…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2The ordinal graph partition relation is monotone in its graph target: if H occurs as a non-induced subgraph-c…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Every graph on at most aleph-one vertices containing neither a four-clique nor a countably infinite complete…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

5) V2 In a graph with no countably infinite complete bipartite subgraph, every countably infinite vertex set has only finitely many common neighbors.

kernel-checked, filed Tue Aug 25 2026 10:35:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All simple graphs and all vertex subsets of cardinality aleph-zero, under the exact no-K_{aleph0,aleph0} hypothesis from Erdős 597.

4) V2 The Erdős 597 partition relation holds for every graph target with at most two vertices.

kernel-checked, filed Tue Aug 25 2026 10:35:20 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All simple graph targets on a type of cardinality at most two, for the fixed omega-one-squared source and omega-one-times-omega red target.

3) V2 For a zero red target, every graph-arrow relation holds because the empty vertex set is red homogeneous and has order type zero.

kernel-checked, filed Tue Aug 25 2026 10:16:41 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All simple graph targets over the fixed omega-one-squared source, with red target order type zero.

2) V2 The ordinal graph partition relation is monotone in its graph target: if H occurs as a non-induced subgraph-copy of G, then the relation for G implies the relation for H.

kernel-checked, filed Tue Aug 25 2026 10:15:15 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All ordinals alpha and beta and all simple graphs H and G with a non-induced copy of H in G.

1) V1 Every graph on at most aleph-one vertices containing neither a four-clique nor a countably infinite complete bipartite subgraph satisfies the red-blue partition relation from omega-one squared to either a red copy of omega-one times omega or a blue copy of the graph.

open, filed Tue Aug 25 2026 10:10:55 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The first general question is posed; the separately printed finite-G question is a special case. The arrow is inlined transparently: false/red yields a subset of inherited order type omega-one times omega; true/blue yields an injective, not necessarily induced graph copy. K_{aleph0,aleph0} means two disjoint sets of cardinality aleph-zero with every cross-edge. No unopened source citation remains.

Scope. All simple graphs of cardinality at most aleph-one with no K4 and no K_{aleph0,aleph0}; arbitrary symmetric red-blue colorings of unordered pairs of omega-one squared.

E597:Erdős Problem #597 “Does the omega-one-squared graph partition relation hold?”

E597: “Does the omega-one-squared graph partition relation hold?”