PartitionRelation.lean`

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1 declarations, 0 contain sorry, 0 declared as axioms

Cardinal partition relation

This file defines Combinatorics.cardinalPartitionRel, the multicolor $r$-uniform partition arrow

$$\mu \to (\nu_\alpha)_{\alpha < \gamma}^r$$

between an infinite "source" cardinal $\mu$, a uniformity $r$, an ordinal-indexed family of "target" cardinals $\nu_\alpha$, and a coloring of $r$-element subsets. This is the standard notation in infinitary combinatorics and is reused by several Erdős–Rado-style problems.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def Combinatorics.cardinalPartitionRel cardinalPartitionRel μ r γ ν asserts the multicolor line 56 source
Source documentation
cardinalPartitionRel μ r γ ν asserts the multicolor $r$-uniform partition relation $$\mu \to (\nu_\alpha)_{\alpha < \gamma}^r.$$
It states: for every type A with #A = μ and every coloring col of the $r$-element finite subsets of A by γ.ToType (the colors indexed by ordinals less than γ), there exists a color i : γ.ToType and a homogeneous subset H : Set A with #H = ν i such that every $r$-element subset of H receives color i.
When γ = 2 and r = 2 this reduces to the classical binary partition relation $\mu \to (\nu_0, \nu_1)^2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Cardinal, Nat, Ordinal, Ordinal.ToType
Proof uses: Eq, Cardinal.mk, Subtype, Finset, Finset.card, Exists, Set, And, LE.le, Subtype.mk
4 instances, coercions and auxiliary constants left out.
```
def cardinalPartitionRel (μ : Cardinal.{u}) (r : ℕ) (γ : Ordinal.{u})
    (ν : γ.ToType → Cardinal.{u}) : Prop :=
…
```