not_cliqueFree_of_infinite — Mathlib

English

If the ENat-card of ι bounds n, the complete multipartite graph is not clique-free.

Русский

Если ENat-кардинал ι ограничивает n, completeMultipartiteGraph не clique-free.

LaTeX

$$$ \text{not_cliqueFree_of_le_enatCard} (f) : n ≤ ENat.card ι \Rightarrow \neg (\mathrm{completeMultipartiteGraph}(V)).IsNClique(n). $$$

Lean4

theorem not_cliqueFree_of_infinite [Infinite ι] (f : ∀ (i : ι), V i) : ¬(completeMultipartiteGraph V).CliqueFree n :=
  fun hf ↦
  not_cliqueFree_of_top_embedding
    (topEmbedding V f |>.comp <| Embedding.completeGraph <| Fin.valEmbedding.trans <| Infinite.natEmbedding ι) hf

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