From ca455ab13373f5cf94e8992742c5cb3f5e26a080 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Ya=C3=ABl=20Dillies?= <yael.dillies@gmail.com>
Date: Sun, 18 Jun 2023 09:17:58 +0000
Subject: [PATCH] feat(combinatorics/simple_graph): Construct a tripartite
 graph from its triangles

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 .../simple_graph/triangle/tripartite.lean     | 264 ++++++++++++++++++
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 create mode 100644 src/combinatorics/simple_graph/triangle/tripartite.lean

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+/-
+Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies, Bhavik Mehta
+-/
+import combinatorics.simple_graph.triangle.basic
+
+/-!
+# Construct a tripartite graph from its triangles
+
+This file contains the construction of a simple graph on `α ⊕ β ⊕ γ` from a list of triangles
+`(a, b, c)` (with `a` in the first summand, `b` in the second, `c` in the third).
+
+We call
+* `t : finset (α × β × γ)` the set of *triangle indices* (its elements are not triangles within the
+  graph but instead index them).
+* *explicit* a triangle of the constructed graph coming from a triangle index.
+* *accidental* a triangle of the constructed graph not coming from a triangle.
+
+The two important properties of this construction are:
+* `simple_graph.tripartite_from_triangles.explicit_disjoint`: Whether the explicit triangles are
+  edge-disjoint.
+* `simple_graph.tripartite_from_triangles.no_accidental`: Whether all triangles are explicit.
+
+This construction shows up unrelatingly twice in the theory of Roth numbers:
+* The lower bound of the Ruzsa-Szemerédi problem: From a Salem-Spencer set `s` on a commutative ring
+  `α` (in which `2` is invertible), we build a locally linear graph on `α ⊕ α ⊕ α`. The triangle
+  indices are `(x, x + a, x + 2 * a)` for `x` any element and `a ∈ s`. The explicit triangles are
+  edge-disjoint and accidental triangles correspond to arithmetic progressions of length 3 in `s`.
+* The proof of the corners theorem from the triangle removal lemma. For a subset `s` of the `n × n`
+  grid, we construct a tripartite graph whose vertices are the horizontal, vertical and diagonal
+  lines in the grid. The explicit triangles are `(h, v, d)` where `h`, `v`, `d` are horizontal,
+  vertical, diagonal lines that intersect in an element of `s`. The explicit triangles are
+  edge-disjoint. Accidental triangles correspond to non-trivial corners.
+
+# Main declarations
+
+* `simple_graph.tripartite_from_triangles.graph`: The tripartite simple graph generated by the given
+  triangle indices.
+* `simple_graph.tripartite_from_triangles.explicit_disjoint`: Predicate on triangle indices for the
+  constructed graph to have edge-disjoint explicit triangles.
+* `simple_graph.tripartite_from_triangles.no_accidental`: Predicate on triangle indices for there to
+  be no accidental triangles.
+* `simple_graph.tripartite_from_triangles.locally_linear`: If both predicates hold, the constructed
+  graph is locally linear.
+-/
+
+open finset function sum3
+
+variables {α β γ 𝕜 : Type*} [linear_ordered_field 𝕜] {t : finset (α × β × γ)} {a a' : α} {b b' : β}
+  {c c' : γ} {x : α × β × γ} {ε : 𝕜}
+
+namespace simple_graph
+namespace tripartite_from_triangles
+
+/-- The underlying relation of the tripartite-from-triangles graph. Two vertices are related iff
+there exists a triangle index containing them both. -/
+@[mk_iff] inductive rel (t : finset (α × β × γ)) : α ⊕ β ⊕ γ → α ⊕ β ⊕ γ → Prop
+| in₀₁ ⦃a b c⦄ : (a, b, c) ∈ t → rel (in₀ a) (in₁ b)
+| in₁₀ ⦃a b c⦄ : (a, b, c) ∈ t → rel (in₁ b) (in₀ a)
+| in₀₂ ⦃a b c⦄ : (a, b, c) ∈ t → rel (in₀ a) (in₂ c)
+| in₂₀ ⦃a b c⦄ : (a, b, c) ∈ t → rel (in₂ c) (in₀ a)
+| in₁₂ ⦃a b c⦄ : (a, b, c) ∈ t → rel (in₁ b) (in₂ c)
+| in₂₁ ⦃a b c⦄ : (a, b, c) ∈ t → rel (in₂ c) (in₁ b)
+
+open rel
+
+lemma rel_irrefl : ∀ x, ¬ rel t x x.
+lemma rel_symm : symmetric (rel t) := λ x y h, by cases h; constructor; assumption
+
+/-- The tripartite-from-triangles graph. Two vertices are related iff there exists a triangle index
+containing them both. -/
+def graph (t : finset (α × β × γ)) : simple_graph (α ⊕ β ⊕ γ) := ⟨rel t, rel_symm, rel_irrefl⟩
+
+namespace graph
+
+@[simp] lemma not_in₀₀ : ¬ (graph t).adj (in₀ a) (in₀ a') .
+@[simp] lemma not_in₁₁ : ¬ (graph t).adj (in₁ b) (in₁ b') .
+@[simp] lemma not_in₂₂ : ¬ (graph t).adj (in₂ c) (in₂ c') .
+
+@[simp] lemma in₀₁_iff : (graph t).adj (in₀ a) (in₁ b) ↔ ∃ c, (a, b, c) ∈ t :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›⟩ }, λ ⟨_, h⟩, in₀₁ h⟩
+@[simp] lemma in₁₀_iff : (graph t).adj (in₁ b) (in₀ a) ↔ ∃ c, (a, b, c) ∈ t :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›⟩ }, λ ⟨_, h⟩, in₁₀ h⟩
+@[simp] lemma in₀₂_iff : (graph t).adj (in₀ a) (in₂ c) ↔ ∃ b, (a, b, c) ∈ t :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›⟩ }, λ ⟨_, h⟩, in₀₂ h⟩
+@[simp] lemma in₂₀_iff : (graph t).adj (in₂ c) (in₀ a) ↔ ∃ b, (a, b, c) ∈ t :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›⟩ }, λ ⟨_, h⟩, in₂₀ h⟩
+@[simp] lemma in₁₂_iff : (graph t).adj (in₁ b) (in₂ c) ↔ ∃ a, (a, b, c) ∈ t :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›⟩ }, λ ⟨_, h⟩, in₁₂ h⟩
+@[simp] lemma in₂₁_iff : (graph t).adj (in₂ c) (in₁ b) ↔ ∃ a, (a, b, c) ∈ t :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›⟩ }, λ ⟨_, h⟩, in₂₁ h⟩
+
+lemma in₀₁_iff' :
+  (graph t).adj (in₀ a) (in₁ b) ↔ ∃ (x : α × β × γ) (hx : x ∈ t), x.1 = a ∧ x.2.1 = b :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›, rfl, rfl⟩ },
+  by { rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩, constructor, assumption }⟩
+lemma in₁₀_iff' :
+  (graph t).adj (in₁ b) (in₀ a) ↔ ∃ (x : α × β × γ) (hx : x ∈ t), x.2.1 = b ∧ x.1 = a :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›, rfl, rfl⟩ },
+  by { rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩, constructor, assumption }⟩
+lemma in₀₂_iff' :
+  (graph t).adj (in₀ a) (in₂ c) ↔ ∃ (x : α × β × γ) (hx : x ∈ t), x.1 = a ∧ x.2.2 = c :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›, rfl, rfl⟩ },
+  by { rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩, constructor, assumption }⟩
+lemma in₂₀_iff' :
+  (graph t).adj (in₂ c) (in₀ a) ↔ ∃ (x : α × β × γ) (hx : x ∈ t), x.2.2 = c ∧ x.1 = a :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›, rfl, rfl⟩ },
+  by { rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩, constructor, assumption }⟩
+lemma in₁₂_iff' :
+  (graph t).adj (in₁ b) (in₂ c) ↔ ∃ (x : α × β × γ) (hx : x ∈ t), x.2.1 = b ∧ x.2.2 = c :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›, rfl, rfl⟩ },
+  by { rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩, constructor, assumption }⟩
+lemma in₂₁_iff' :
+  (graph t).adj (in₂ c) (in₁ b) ↔ ∃ (x : α × β × γ) (hx : x ∈ t), x.2.2 = c ∧ x.2.1 = b :=
+⟨by { rintro ⟨⟩, exact ⟨_, ‹_›, rfl, rfl⟩ },
+  by { rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩, constructor, assumption }⟩
+
+end graph
+
+open graph
+
+/-- Predicate on the triangle indices for the explicit triangles to be edge-disjoint. -/
+class explicit_disjoint (t : finset (α × β × γ)) : Prop :=
+(inj₀ : ∀ ⦃a b c a'⦄, (a, b, c) ∈ t → (a', b, c) ∈ t → a = a')
+(inj₁ : ∀ ⦃a b c b'⦄, (a, b, c) ∈ t → (a, b', c) ∈ t → b = b')
+(inj₂ : ∀ ⦃a b c c'⦄, (a, b, c) ∈ t → (a, b, c') ∈ t → c = c')
+
+/-- Predicate on the triangle indices for there to be no accidental triangle.
+
+Note that we cheat a bit, since the exact translation of this informal description would have
+`(a', b', c') ∈ s` as a conclusion rather than `a = a' ∨ b = b' ∨ c = c'`. Those conditions are
+equivalent when the explicit triangles are edge-disjoint (which is the case we care about). -/
+class no_accidental (t : finset (α × β × γ)) : Prop :=
+(wow : ∀ ⦃a a' b b' c c'⦄, (a', b, c) ∈ t → (a, b', c) ∈ t → (a, b, c') ∈ t →
+  a = a' ∨ b = b' ∨ c = c')
+
+section decidable_eq
+variables [decidable_eq α] [decidable_eq β] [decidable_eq γ]
+
+instance : decidable_rel (graph t).adj
+| (in₀ a) (in₀ a') := decidable.is_false not_in₀₀
+| (in₀ a) (in₁ b') := decidable_of_iff' _ in₀₁_iff'
+| (in₀ a) (in₂ c') := decidable_of_iff' _ in₀₂_iff'
+| (in₁ b) (in₀ a') := decidable_of_iff' _ in₁₀_iff'
+| (in₁ b) (in₁ b') := decidable.is_false not_in₁₁
+| (in₁ b) (in₂ b') := decidable_of_iff' _ in₁₂_iff'
+| (in₂ c) (in₀ a') := decidable_of_iff' _ in₂₀_iff'
+| (in₂ c) (in₁ b') := decidable_of_iff' _ in₂₁_iff'
+| (in₂ c) (in₂ b') := decidable.is_false not_in₂₂
+
+/-- This lemma reorders the elements of a triangle in the tripartite graph. It turns a triangle
+`{x, y, z}` into a triangle `{a, b, c}` where `a : α `, `b : β`, `c : γ`. -/
+ lemma graph_triple ⦃x y z⦄ :
+  (graph t).adj x y → (graph t).adj x z → (graph t).adj y z → ∃ a b c,
+    ({in₀ a, in₁ b, in₂ c} : finset (α ⊕ β ⊕ γ)) = {x, y, z} ∧ (graph t).adj (in₀ a) (in₁ b) ∧
+      (graph t).adj (in₀ a) (in₂ c) ∧ (graph t).adj (in₁ b) (in₂ c) :=
+by rintro (_ | _ | _) (_ | _ | _) (_ | _ | _); refine ⟨_, _, _, by ext; simp only
+  [finset.mem_insert, finset.mem_singleton]; try { tauto }, _, _, _⟩; constructor; assumption
+
+/-- The map that turns a triangle index into an explicit triangle. -/
+@[simps] def to_triangle : α × β × γ ↪ finset (α ⊕ β ⊕ γ) :=
+{ to_fun := λ x, {in₀ x.1, in₁ x.2.1, in₂ x.2.2},
+  inj' := λ ⟨a, b, c⟩ ⟨a', b', c'⟩, by simpa only [finset.subset.antisymm_iff, finset.subset_iff,
+    mem_insert, mem_singleton, forall_eq_or_imp, forall_eq, prod.mk.inj_iff, or_false, false_or,
+    in₀, in₁, in₂, sum.inl.inj_eq, sum.inr.inj_eq] using and.left }
+
+lemma to_triangle_is_3_clique (hx : x ∈ t) : (graph t).is_n_clique 3 (to_triangle x) :=
+begin
+  rcases x with ⟨a, b, c⟩,
+  simp only [to_triangle_apply, is_3_clique_triple_iff, in₀₁_iff, in₀₂_iff, in₁₂_iff],
+  exact ⟨⟨_, hx⟩, ⟨_, hx⟩, _, hx⟩,
+end
+
+lemma exists_mem_to_triangle {x y : α ⊕ β ⊕ γ} (hxy : (graph t).adj x y) :
+  ∃ z ∈ t, x ∈ to_triangle z ∧ y ∈ to_triangle z :=
+by cases hxy; exact ⟨_, ‹_›, by simp⟩
+
+lemma is_3_clique_iff [no_accidental t] {s : finset (α ⊕ β ⊕ γ)} :
+  (graph t).is_n_clique 3 s ↔ ∃ x, x ∈ t ∧ to_triangle x = s :=
+begin
+  refine ⟨λ h, _, _⟩,
+  { rw is_3_clique_iff at h,
+    obtain ⟨x, y, z, hxy, hxz, hyz, rfl⟩ := h,
+    obtain ⟨a, b, c, habc, hab, hac, hbc⟩ := graph_triple hxy hxz hyz,
+    refine ⟨(a, b, c), _, habc⟩,
+    obtain ⟨c', hc'⟩ := in₀₁_iff.1 hab,
+    obtain ⟨b', hb'⟩ := in₀₂_iff.1 hac,
+    obtain ⟨a', ha'⟩ := in₁₂_iff.1 hbc,
+    obtain (rfl | rfl | rfl) := no_accidental.wow ha' hb' hc'; assumption },
+  { rintro ⟨x, hx, rfl⟩,
+    exact to_triangle_is_3_clique hx }
+end
+
+lemma to_triangle_surj_on [no_accidental t] :
+  (t : set (α × β × γ)).surj_on to_triangle ((graph t).clique_set 3) :=
+λ s, is_3_clique_iff.1
+
+variables (t)
+
+lemma map_to_triangle_disjoint [explicit_disjoint t] :
+  (t.map to_triangle : set (finset (α ⊕ β ⊕ γ))).pairwise $
+    λ x y, (x ∩ y : set (α ⊕ β ⊕ γ)).subsingleton :=
+begin
+  intro,
+  simp only [finset.coe_map, set.mem_image, finset.mem_coe, prod.exists, ne.def,
+    forall_exists_index, and_imp],
+  rintro a b c habc rfl e x y z hxyz rfl h',
+  have := ne_of_apply_ne _ h',
+  simp only [ne.def, prod.mk.inj_iff, not_and] at this,
+  simp only [to_triangle_apply, in₀, in₁, in₂, set.mem_inter_iff, mem_insert, mem_singleton,
+    mem_coe, and_imp, sum.forall, or_false, forall_eq, false_or, eq_self_iff_true, implies_true_iff,
+    true_and, and_true, set.subsingleton],
+  suffices : ¬ (a = x ∧ b = y) ∧ ¬ (a = x ∧ c = z) ∧ ¬ (b = y ∧ c = z),
+  { tauto },
+  refine ⟨_, _, _⟩,
+  { rintro ⟨rfl, rfl⟩,
+    exact this rfl rfl (explicit_disjoint.inj₂ habc hxyz) },
+  { rintro ⟨rfl, rfl⟩,
+    exact this rfl (explicit_disjoint.inj₁ habc hxyz) rfl },
+  { rintro ⟨rfl, rfl⟩,
+    exact this (explicit_disjoint.inj₀ habc hxyz) rfl rfl }
+end
+
+lemma clique_set_eq_image [no_accidental t] : (graph t).clique_set 3 = to_triangle '' t :=
+by ext; exact is_3_clique_iff
+
+section fintype
+variables [fintype α] [fintype β] [fintype γ]
+
+lemma clique_finset_eq_image [no_accidental t] : (graph t).clique_finset 3 = t.image to_triangle :=
+coe_injective $ by { push_cast, exact clique_set_eq_image _ }
+
+lemma clique_finset_eq_map [no_accidental t] : (graph t).clique_finset 3 = t.map to_triangle :=
+by simp [clique_finset_eq_image, map_eq_image]
+
+@[simp] lemma card_triangles [no_accidental t] : ((graph t).clique_finset 3).card = t.card :=
+by rw [clique_finset_eq_map, card_map]
+
+lemma far_from_triangle_free [explicit_disjoint t] {ε : 𝕜}
+  (ht : ε * ((fintype.card α + fintype.card β + fintype.card γ) ^ 2 : ℕ) ≤ t.card) :
+  (graph t).far_from_triangle_free ε :=
+far_from_triangle_free_of_disjoint_triangles (t.map to_triangle)
+  (map_subset_iff_subset_preimage.2 $ λ x hx, by simpa using to_triangle_is_3_clique hx)
+  (map_to_triangle_disjoint t) $  by simpa [add_assoc] using ht
+
+end fintype
+end decidable_eq
+
+variables (t)
+
+lemma locally_linear [explicit_disjoint t] [no_accidental t] : (graph t).locally_linear :=
+begin
+  classical,
+  refine ⟨_, λ x y hxy, _⟩,
+  { unfold edge_disjoint_triangles,
+    convert map_to_triangle_disjoint t,
+    rw [clique_set_eq_image, coe_map] },
+  { obtain ⟨z, hz, hxy⟩ := exists_mem_to_triangle hxy,
+    exact ⟨_, to_triangle_is_3_clique hz, hxy⟩ }
+end
+
+end tripartite_from_triangles
+end simple_graph