feat(ErdosProblems): 13

FormalConjectures/ErdosProblems/13.lean

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+/-
+Copyright 2025 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 13
+
+*Reference:* [erdosproblems.com/13](https://www.erdosproblems.com/13)
+-/
+
+open Finset Nat
+
+namespace Erdos13
+
+/--
+A finite set of naturals `A` is said to be forbidden-triple-free if for all `a, b, c ∈ A`,
+if `a < min(b, c)` then `a` does not divide `b + c`.
+-/
+def IsForbiddenTripleFree (A : Finset ℕ) : Prop :=
+  ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, a < min b c → ¬ (a ∣ b + c)
+
+/--
+If $A \subseteq \{1, ..., N\}$ is a set with no $a, b, c \in A$ such that $a | (b+c)$ and
+$a < \min(b,c)$, then $|A| \le N/3 + O(1)$. This has been solved by Bedert [Be23].
+
+[Be23] Bedert, B., _On a problem of Erdős and Sárközy about sequences with no term dividing
+the sum of two larger terms_. arXiv:2301.07065 (2023).
+-/
+@[category research solved, AMS 5 11]
+theorem erdos_13 : ∃ C : ℝ, ∀ N : ℕ, ∀ A ⊆ Icc 1 N, IsForbiddenTripleFree A →
+    (A.card : ℝ) ≤ (N : ℝ) / 3 + C := by
+  sorry
+
+/--
+A general version asks, for a fixed $r \in \mathbb{N}$, if a set
+$A \subseteq \{1, ..., N\}$ has no $a \in A$ and $b_1, ..., b_r \in A$ such that
+$a | (b_1 + ... + b_r)$ and $a < \min(b_1, ..., b_r)$, then is it true that
+$|A| \le N/(r+1) + O(1)$?
+-/
+@[category research open, AMS 5 11]
+theorem erdos_13_general (r : ℕ) : ∃ C : ℝ, ∀ N : ℕ,
+    ∀ A ⊆ Icc 1 N,
+    (∀ a ∈ A, ∀ (b : Fin r → ℕ), (∀ i, b i ∈ A) → (∀ i, a < b i) →
+      ¬ (a ∣ ∑ i, b i)) →
+    (A.card : ℝ) ≤ (N : ℝ) / (r + 1) + C := by
+  sorry
+
+end Erdos13