feat(ErdosProblems/1105): anti-Ramsey numbers for cycles and paths

FormalConjectures/ErdosProblems/1105.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
+
+/-!
+# Erdős Problem 1105
+
+The anti-Ramsey number AR(n, G) is the maximum possible number of colours in which the edges of
+Kₙ can be coloured without creating a rainbow copy of G (i.e. one in which all edges have
+different colours).
+
+## Conjecture for Cycles
+
+Let Cₖ be the cycle on k vertices. Is it true that
+  AR(n, Cₖ) = ((k-2)/2 + 1/(k-1)) n + O(1)?
+
+## Conjecture for Paths
+
+Let Pₖ be the path on k vertices and ℓ = ⌊(k-1)/2⌋. If n ≥ k ≥ 5 then is AR(n, Pₖ) equal to
+  max(C(k-2, 2) + 1, C(ℓ-1, 2) + (ℓ-1)(n-ℓ+1) + ε)
+where ε = 1 if k is odd and ε = 2 otherwise?
+
+A conjecture of Erdős, Simonovits, and Sós [ESS75], who gave a simple proof that AR(n, C₃) = n-1.
+Simonovits and Sós [SiSo84] published a proof that the claimed formula for AR(n, Pₖ) is true
+for n ≥ ck² for some constant c > 0.
+A proof of the formula for AR(n, Pₖ) for all n ≥ k ≥ 5 has been announced by Yuan [Yu21].
+
+*Reference:* [erdosproblems.com/1105](https://www.erdosproblems.com/1105)
+-/
+
+namespace Erdos1105
+
+open SimpleGraph
+
+/-- The path graph on `k` vertices: vertices are `Fin k` and vertex `i` is adjacent to `i+1`. -/
+def pathGraph (k : ℕ) : SimpleGraph (Fin k) :=
+  SimpleGraph.fromRel fun i j => i.val + 1 = j.val
+
+/-- An edge coloring of a simple graph `G` with color type `α`. -/
+def EdgeColoring (G : SimpleGraph V) (α : Type*) := G.edgeSet → α
+
+/-- A subgraph `H` of a graph with edge coloring `c` is rainbow if all edges of `H` have distinct
+colors. -/
+def IsRainbow {V : Type*} {G : SimpleGraph V} (c : EdgeColoring G α) (H : G.Subgraph) : Prop :=
+  Function.Injective fun e : H.edgeSet => c ⟨e.val, H.edgeSet_subset e.property⟩
+
+/-- A graph `G` contains a rainbow copy of `H` if there is a subgraph of `G` that is isomorphic
+to `H` and is rainbow under the edge coloring `c`. -/
+def HasRainbowCopy {V W : Type*} {G : SimpleGraph V} (c : EdgeColoring G α) (H : SimpleGraph W) :
+    Prop :=
+  ∃ (S : G.Subgraph), H ⊑ S.coe ∧ IsRainbow c S
+
+/-- An edge coloring of `Kₙ` with `m` colors that avoids rainbow copies of `H`. -/
+def IsAntiRamseyColoring (n m : ℕ) (H : SimpleGraph W) : Prop :=
+  ∃ (c : EdgeColoring (completeGraph (Fin n)) (Fin m)), ¬HasRainbowCopy c H
+
+/-- The anti-Ramsey number AR(n, H) is the maximum number of colors in which the edges of Kₙ
+can be colored without creating a rainbow copy of H. -/
+noncomputable def antiRamseyNumber (n : ℕ) (H : SimpleGraph W) : ℕ :=
+  sSup {m : ℕ | IsAntiRamseyColoring n m H}
+
+/-- The conjectured value for the anti-Ramsey number of cycles.
+  AR(n, Cₖ) ≈ ((k-2)/2 + 1/(k-1)) · n -/
+noncomputable def conjecturedARCycle (n k : ℕ) : ℝ :=
+  ((k - 2 : ℝ) / 2 + 1 / (k - 1)) * n
+
+/-- The conjectured value for the anti-Ramsey number of paths.
+  AR(n, Pₖ) = max(C(k-2, 2) + 1, C(ℓ-1, 2) + (ℓ-1)(n-ℓ+1) + ε)
+where ℓ = ⌊(k-1)/2⌋ and ε = 1 if k is odd, ε = 2 otherwise. -/
+noncomputable def conjecturedARPath (n k : ℕ) : ℕ :=
+  let ell := (k - 1) / 2  -- ℓ = ⌊(k-1)/2⌋
+  let eps := if k % 2 = 1 then 1 else 2  -- ε = 1 if k odd, 2 otherwise
+  max (Nat.choose (k - 2) 2 + 1)
+      (Nat.choose (ell - 1) 2 + (ell - 1) * (n - ell + 1) + eps)
+
+/--
+The anti-Ramsey number for cycles satisfies
+  AR(n, Cₖ) = ((k-2)/2 + 1/(k-1)) · n + O(1).
+This is a conjecture of Erdős, Simonovits, and Sós [ESS75].
+-/
+@[category research open, AMS 5]
+theorem erdos_1105_cycles (k : ℕ) (hk : 3 ≤ k) :
+    ∃ C : ℝ, ∀ n : ℕ, |antiRamseyNumber n (cycleGraph k) - conjecturedARCycle n k| ≤ C := by
+  sorry
+
+/--
+The anti-Ramsey number for paths satisfies
+  AR(n, Pₖ) = max(C(k-2,2) + 1, C(ℓ-1,2) + (ℓ-1)(n-ℓ+1) + ε)
+where ℓ = ⌊(k-1)/2⌋ and ε = 1 if k is odd and ε = 2 otherwise.
+This is a conjecture of Erdős, Simonovits, and Sós [ESS75].
+The case n ≥ k ≥ 5 has been announced as proven by Yuan [Yu21].
+-/
+@[category research open, AMS 5]
+theorem erdos_1105.variants.paths (n k : ℕ) (hn : k ≤ n) (hk : 5 ≤ k) :
+    antiRamseyNumber n (pathGraph k) = conjecturedARPath n k := by
+  sorry
+
+/--
+Known result: AR(n, C₃) = n - 1.
+Proved by Erdős, Simonovits, and Sós [ESS75].
+-/
+@[category research solved, AMS 5]
+theorem erdos_1105.variants.triangles (n : ℕ) (hn : 3 ≤ n) :
+    antiRamseyNumber n (cycleGraph 3) = n - 1 := by
+  sorry
+
+end Erdos1105
+
+