feat(Paper): three-sunflower-free set systems

FormalConjectures/Paper/ThreeSunflowerFreeSetSystems.lean

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+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+import FormalConjecturesForMathlib.Combinatorics.SetFamily.Sunflower
+
+/-!
+# Three-sunflower-free set systems with bounded pairwise intersections
+
+*References:*
+* [Three-sunflower-free set systems with bounded pairwise intersections]
+  (https://doi.org/10.5281/zenodo.20693260), by *Cody Mitchell* (2026).
+* [sunflower-lean: paper-v2](https://doi.org/10.5281/zenodo.20693191),
+  companion Lean 4 formalization, release `paper-v2`.
+
+For `ℓ > t ≥ 1`, the paper studies the maximum size `M₃(ℓ,t)` of a
+family of distinct `ℓ`-sets with pairwise intersections of size at most `t`
+and no three-sunflower. The empty-core case is included, so three pairwise
+disjoint sets form a three-sunflower. The variant `I₃(ℓ,t)` imposes the
+additional condition that the family is intersecting.
+-/
+
+open Filter
+
+namespace ThreeSunflowerFreeSetSystems
+
+variable {α : Type}
+
+/-- A family is `ℓ`-uniform if every member is finite of cardinality `ℓ`. -/
+def IsUniform (ℓ : ℕ) (F : Set (Set α)) : Prop :=
+  ∀ A ∈ F, A.Finite ∧ A.ncard = ℓ
+
+/-- Every two distinct members of `F` have intersection size at most `t`. -/
+def HasPairwiseIntersectionsAtMost (t : ℕ) (F : Set (Set α)) : Prop :=
+  ∀ A ∈ F, ∀ B ∈ F, A ≠ B → (A ∩ B).ncard ≤ t
+
+/-- The family `F` contains no three-member sunflower. -/
+def ThreeSunflowerFree (F : Set (Set α)) : Prop :=
+  ¬ ∃ S : Set (Set α), S ⊆ F ∧ S.ncard = 3 ∧ IsSunflower S
+
+/--
+An admissible family for `M₃(ℓ,t)`: a distinct family of `ℓ`-sets, all
+pairwise intersections have size at most `t`, and no three members form a
+sunflower.
+-/
+def M3Admissible (ℓ t : ℕ) (F : Set (Set α)) : Prop :=
+  IsUniform ℓ F ∧ HasPairwiseIntersectionsAtMost t F ∧ ThreeSunflowerFree F
+
+/--
+An admissible family for `I₃(ℓ,t)`: an admissible family for `M₃(ℓ,t)` with
+no disjoint pair.
+-/
+def I3Admissible (ℓ t : ℕ) (F : Set (Set α)) : Prop :=
+  M3Admissible ℓ t F ∧ ∀ A ∈ F, ∀ B ∈ F, A ≠ B → (A ∩ B).Nonempty
+
+/--
+The extremal number `M₃(ℓ,t)`: the largest size, over finite ground sets, of
+a three-sunflower-free `ℓ`-uniform family whose pairwise intersections have
+size at most `t`.
+-/
+noncomputable def M3 (ℓ t : ℕ) : ℕ :=
+  sSup {m : ℕ | ∃ (α : Type) (_ : Fintype α) (F : Set (Set α)),
+    M3Admissible ℓ t F ∧ F.ncard = m}
+
+/--
+The intersecting extremal number `I₃(ℓ,t)`: the same maximum as `M₃(ℓ,t)`,
+with the additional restriction that the family has no disjoint pair.
+-/
+noncomputable def I3 (ℓ t : ℕ) : ℕ :=
+  sSup {m : ℕ | ∃ (α : Type) (_ : Fintype α) (F : Set (Set α)),
+    I3Admissible ℓ t F ∧ F.ncard = m}
+
+/--
+The restricted-intersection three-sunflower threshold: the least `N` such
+that every `n`-uniform family with pairwise intersections of size at most `t`
+and at least `N` members contains a three-sunflower.
+-/
+noncomputable def restrictedThreshold (n t : ℕ) : ℕ :=
+  sInf {N : ℕ | ∀ {α : Type}, ∀ F : Set (Set α),
+    IsUniform n F → HasPairwiseIntersectionsAtMost t F → N ≤ F.ncard →
+      ∃ S ⊆ F, S.ncard = 3 ∧ IsSunflower S}
+
+/-- A natural number is a prime power. -/
+def IsPrimePower (q : ℕ) : Prop :=
+  ∃ p a : ℕ, p.Prime ∧ 0 < a ∧ q = p ^ a
+
+/--
+The two-copy decomposition supplied by the `t = 1` classification: the family
+splits into two disjoint intersecting extremal pieces, and every cross pair is
+disjoint.
+-/
+def HasTwoDisjointT1ExtremalPieces (ℓ : ℕ) (F : Set (Set α)) : Prop :=
+  ∃ G H : Set (Set α), G ⊆ F ∧ H ⊆ F ∧ F = G ∪ H ∧ Disjoint G H ∧
+    I3Admissible ℓ 1 G ∧ I3Admissible ℓ 1 H ∧ G.ncard = ℓ + 1 ∧ H.ncard = ℓ + 1 ∧
+    ∀ A ∈ G, ∀ B ∈ H, Disjoint A B
+
+/--
+The exact `t = 1` values from Mitchell's paper: for every `ℓ ≥ 2`,
+`I₃(ℓ,1) = ℓ + 1` and `M₃(ℓ,1) = 2ℓ + 2`.
+-/
+@[category research solved, AMS 5,
+  formal_proof using lean4 at "https://github.com/SproutSeeds/sunflower-lean/tree/paper-v2"]
+theorem m3_t1_exact (ℓ : ℕ) (hℓ : 2 ≤ ℓ) :
+    I3 ℓ 1 = ℓ + 1 ∧ M3 ℓ 1 = 2 * ℓ + 2 := by
+  sorry
+
+/--
+The `t = 1` extremal classification implies that every extremal `M₃(ℓ,1)`
+family splits into two disjoint intersecting extremal pieces. The companion
+Lean development proves the sharper vertex-star incidence classification of
+those pieces.
+-/
+@[category research solved, AMS 5,
+  formal_proof using lean4 at "https://github.com/SproutSeeds/sunflower-lean/tree/paper-v2"]
+theorem m3_t1_extremal_decomposition (ℓ : ℕ) (hℓ : 2 ≤ ℓ) (F : Set (Set α))
+    (hF : M3Admissible ℓ 1 F) (hcard : F.ncard = M3 ℓ 1) :
+    HasTwoDisjointT1ExtremalPieces ℓ F := by
+  sorry
+
+/--
+The sharp counting upper bound at `t = 2`: for every `ℓ ≥ 3`,
+`M₃(ℓ,2) ≤ 3ℓ² - ℓ + 2`.
+-/
+@[category research solved, AMS 5,
+  formal_proof using lean4 at "https://github.com/SproutSeeds/sunflower-lean/tree/paper-v2"]
+theorem m3_t2_upper_bound (ℓ : ℕ) (hℓ : 3 ≤ ℓ) :
+    M3 ℓ 2 ≤ 3 * ℓ ^ 2 - ℓ + 2 := by
+  sorry
+
+/--
+The orthogonal-projective-plane construction gives the lower bound
+`M₃(2q+2,2) ≥ 2(q²+q+1)` for every prime power `q`.
+-/
+@[category research solved, AMS 5,
+  formal_proof using lean4 at "https://github.com/SproutSeeds/sunflower-lean/tree/paper-v2"]
+theorem m3_t2_prime_power_lower_bound (q : ℕ) (hq : IsPrimePower q) :
+    2 * (q ^ 2 + q + 1) ≤ M3 (2 * q + 2) 2 := by
+  sorry
+
+/--
+The paper's unconditional quadratic lower bound at `t = 2`, obtained from
+the prime-power construction by padding and Bertrand's postulate.
+-/
+@[category research solved, AMS 5,
+  formal_proof using lean4 at "https://github.com/SproutSeeds/sunflower-lean/tree/paper-v2"]
+theorem m3_t2_quadratic_lower_bound (ℓ : ℕ) (hℓ : 4 ≤ ℓ) :
+    (ℓ - 2) ^ 2 / 8 ≤ M3 ℓ 2 := by
+  sorry
+
+/--
+Corollary 1.3 of the paper: for bounded pairwise intersections, the
+three-sunflower threshold is one more than the extremal number.
+-/
+@[category research solved, AMS 5]
+theorem restricted_threshold_eq_m3_add_one (n t : ℕ) (htn : t < n) :
+    restrictedThreshold n t = M3 n t + 1 := by
+  sorry
+
+/--
+The disjointness graph of a family is Mantel-tight when its ordered disjoint
+pairs attain the balanced triangle-free extremal count.
+-/
+def HasMantelTightDisjointness (F : Set (Set α)) : Prop :=
+  {p : Set α × Set α | p.1 ∈ F ∧ p.2 ∈ F ∧ p.1 ≠ p.2 ∧ Disjoint p.1 p.2}.ncard =
+    2 * (F.ncard ^ 2 / 4)
+
+/--
+The family `F` splits into two intersecting admissible pieces, with every
+cross pair disjoint.
+-/
+def SplitsIntoTwoIntersectingPieces (ℓ t : ℕ) (F : Set (Set α)) : Prop :=
+  ∃ G H : Set (Set α), G ⊆ F ∧ H ⊆ F ∧ F = G ∪ H ∧ Disjoint G H ∧
+    I3Admissible ℓ t G ∧ I3Admissible ℓ t H ∧
+    ∀ A ∈ G, ∀ B ∈ H, Disjoint A B
+
+/--
+Version 2 structural reduction at `t = 2`: in the Mantel-tight
+disjointness regime, an admissible family is exactly two intersecting
+admissible pieces on disjoint supports.
+-/
+@[category research solved, AMS 5,
+  formal_proof using lean4 at "https://github.com/SproutSeeds/sunflower-lean/tree/paper-v2"]
+theorem m3_t2_mantel_tight_reduction {α : Type} (ℓ : ℕ) (F : Set (Set α))
+    (hF : M3Admissible ℓ 2 F) (hTight : HasMantelTightDisjointness F) :
+    SplitsIntoTwoIntersectingPieces ℓ 2 F := by
+  sorry
+
+/--
+Open exponent problem from the paper: is `M₃(ℓ,t)` quadratically bounded in
+`ℓ` for every fixed `t ≥ 2`?
+-/
+@[category research open, AMS 5]
+theorem m3_fixed_t_quadratic_exponent_problem :
+    answer(sorry) ↔
+      ∀ t : ℕ, 2 ≤ t → ∃ C : ℕ, ∀ ℓ : ℕ, t < ℓ → M3 ℓ t ≤ C * ℓ ^ 2 := by
+  sorry
+
+/--
+Open constant problem at `t = 2`: does the normalized sequence
+`M₃(ℓ,2) / ℓ²` converge?
+-/
+@[category research open, AMS 5]
+theorem m3_t2_constant_problem :
+    answer(sorry) ↔
+      ∃ c : ℝ, Tendsto (fun ℓ : ℕ => (M3 ℓ 2 : ℝ) / (ℓ : ℝ) ^ 2) atTop (nhds c) := by
+  sorry
+
+/--
+Open structural problem at `t = 2`: is two-copy doubling of an optimal
+intersecting family asymptotically optimal up to an additive constant?
+-/
+@[category research open, AMS 5]
+theorem m3_t2_doubling_optimal_problem :
+    answer(sorry) ↔ ∃ C : ℕ, ∀ ℓ : ℕ, 3 ≤ ℓ → M3 ℓ 2 ≤ 2 * I3 ℓ 2 + C := by
+  sorry
+
+end ThreeSunflowerFreeSetSystems