HomDensity.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

Homomorphism density for finite simple graphs

Mathlib does not (as of 2026-04) provide SimpleGraph.homDensity. We define a lightweight version here, specialised to finite vertex types on the host G, and prove the basic homDensity H G ≤ 1 upper bound used as a sanity check in Sidorenko-style arguments.

The extension to infinite hosts uses the same formula with Fintype.card replaced by measure-theoretic integrals; we do not need that generality here.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def SimpleGraph.homCount The number of graph homomorphisms H → G, as a natural number line 41 source
Source documentation
The number of graph homomorphisms H → G, as a natural number. For finite vertex types this is just the cardinality of the Hom type.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat
Proof uses: Fintype.card, SimpleGraph.Hom
1 instances, coercions and auxiliary constants left out.
```
noncomputable def homCount [Fintype V] [Fintype W] [DecidableEq V] [DecidableEq W]
    (H : SimpleGraph V) (G : SimpleGraph W) [DecidableRel H.Adj] [DecidableRel G.Adj] : ℕ :=
…
```
def SimpleGraph.homDensity The homomorphism density t(H, G) of a simple graph H in a simple graph G with finite vertex sets is #Hom(H, G) / |V(G)|^{|V(H)|} line 47 source
Source documentation
The homomorphism density t(H, G) of a simple graph H in a simple graph G with finite vertex sets is #Hom(H, G) / |V(G)|^{|V(H)|}.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Real
Proof uses: HDiv.hDiv, SimpleGraph.homCount, HPow.hPow, Nat, Fintype.card
9 instances, coercions and auxiliary constants left out.
```
noncomputable def homDensity [Fintype V] [Fintype W] [DecidableEq V] [DecidableEq W]
    (H : SimpleGraph V) (G : SimpleGraph W) [DecidableRel H.Adj] [DecidableRel G.Adj] : ℝ :=
…
```
lemma SimpleGraph.homDensity_le_one The homomorphism density t(H, G) is at most 1 line 55 source
Source documentation
The homomorphism density t(H, G) is at most 1.
Proof outline: the underlying function of a H →g G-homomorphism is an element of V → W, which has |W|^|V| elements, so homCount H G ≤ |W|^|V|. Divide.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nonempty, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, Real, SimpleGraph.homDensity, OfNat.ofNat
Proof uses: LT.lt, Nat, Fintype.card, Fintype.card_pos, SimpleGraph.Hom, Fintype.card_le_of_injective, DFunLike.coe_injective, Eq, HPow.hPow, Fintype.card_fun, SimpleGraph.homCount, id, Eq.rec, Iff.mpr, Nat.cast_le, Eq.mpr, Eq.mp, congrArg, Nat.cast_pow, pow_pos and 9 more
72 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.
```
lemma homDensity_le_one {V W : Type*} [Fintype V] [Fintype W] [Nonempty W]
    [DecidableEq V] [DecidableEq W]
    (H : SimpleGraph V) (G : SimpleGraph W)
    [DecidableRel H.Adj] [DecidableRel G.Adj] :
    homDensity H G ≤ 1 := by
…
```
Mentions: SimpleGraph.homDensity