Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Induced removal must permit adding edges as well as deleting them

Statement

For every n≥3 there is an n-vertex graph with an induced empty three-vertex graph that cannot be destroyed by edge deletions, although adding one edge destroys that induced copy.

Facts & Assumptions

Given: An integer n≥3.

[L1]

Induced removal permits changing adjacencies in both directions (Induced graph removal lemma for a fixed graph).

[L2]

The empty graph on three vertices has no edges (Empty and complete graphs, complete bipartite graphs, and the convention that Pn and Cn have n vertices).

[L3]

Labelled induced copies are injective maps preserving edges and nonedges (The induced-embedding count ind⁡H(G)).

Counterexample

technique · direct
1.1

Begin with Kn, choose a triple S, and delete exactly its three internal edges. By [L2] and [L3], S induces the empty three-vertex graph.

givenL2L3choose
2.1

It is the unique unlabelled empty triple: every triple other than S contains a vertex outside S, and that vertex is adjacent to both other vertices.

step 1.1
2.2

Deleting more edges never changes any of the three nonedges within S into an edge, so the induced empty triple on S survives every deletion-only operation.

step 1.1
3.1

Adding any one of the three missing edges within S destroys this copy, and step 2.1 shows the resulting graph has no empty triple. Thus allowing additions, as [L1] does, is indispensable.

step 2.1step 2.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources