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Mathlib.Combinatorics.SimpleGraph.Regularity.Energy

Energy of a partition #

This file defines the energy of a partition.

The energy is the auxiliary quantity that drives the induction process in the proof of Szemerédi's Regularity Lemma. As long as we do not have a suitable equipartition, we will find a new one that has an energy greater than the previous one plus some fixed constant.

References #

[Yaël Dillies, Bhavik Mehta, Formalising Szemerédi’s Regularity Lemma in Lean][srl_itp]

def Finpartition.energy {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α) [DecidableRel G.Adj] :

The energy of a partition, also known as index. Auxiliary quantity for Szemerédi's regularity lemma.

Instances For
    @[simp]
    theorem Finpartition.coe_energy {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G : SimpleGraph α) [DecidableRel G.Adj] {𝕜 : Type u_2} [LinearOrderedField 𝕜] :
    ↑(Finpartition.energy P G) = (Finset.sum (Finset.offDiag P.parts) fun uv => ↑(SimpleGraph.edgeDensity G uv.fst uv.snd) ^ 2) / ↑(Finset.card P.parts) ^ 2