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Lax57.SparseHouseTools

Anticomplete pairs in sparse P5P_5-free graphs

concepts/Lax57/SparseHouseTools.lean · lax-57

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    Theorem

    A 1/321/32-sparse P5P_5-free graph on a set SS of at least two vertices has two anticomplete sets, each of size at least ∣S∣/32|S|/32. This is Lemma 4.4 of Nguyen, Scott, and Seymour.

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    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Anticomplete pairs in sparse P5P_5-free graphs
    6type: theorem
    7---
    8A 1/321/32-sparse P5P_5-free graph on a set SS of at least two vertices
    9has two anticomplete sets, each of size at least ∣S∣/32|S|/32. This is Lemma 4.4
    10of Nguyen, Scott, and Seymour.
    11-/
    12
    13namespace Lax57.SparseHouseTools
    14
    15open Lax57.GraphDefinitions
    16
    17universe u
    18
    19/-- The linear anticomplete-pair lemma for sparse `P5`-free graphs. -/
    20axiom sparse_P5_anticomplete_pair :
    21 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    22 (G : SimpleGraph V) [DecidableRel G.Adj] (S : Finset V),
    23 IsP5Free G → 2 ≤ S.card → ESparse G 32 S →
    24 ∃ B : Blockade (V := V) 2,
    25 B.IsInside S ∧ B.IsAnticomplete G ∧
    26 ∀ i : Fin 2, S.card ≤ 32 * (B.block i).card
    27
    28end Lax57.SparseHouseTools
    29
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