feat(ErdosProblems): 740

FormalConjectures/ErdosProblems/740.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 740: Infinitary version of chromatic number and odd cycles
+
+*Reference:* [erdosproblems.com/740](https://erdosproblems.com/740)
+-/
+
+namespace Erdos740
+
+variable {V : Type*}
+
+/-- A graph avoids odd cycles of length ≤ r if it contains no odd cycles of length at most r -/
+def avoidsOddCyclesOfLength (G : SimpleGraph V) (r : ℕ) : Prop :=
+  ∀ (n : ℕ) (v : V) (c : G.Walk v v), c.length = n → n ≤ r → Odd n → ¬c.IsCycle
+
+/--
+Does every graph $G$ with $\chi(G) = \infty$ contain, for every $r \in \mathbb{N}$, a subgraph
+$H \subseteq G$ such that $\chi(H) = \infty$ and $H$ contains no odd cycle of length $\le r$?
+-/
+@[category research open, AMS 5]
+theorem erdos_740 :
+    answer(sorry) ↔
+      ∀ (r : ℕ) (G : SimpleGraph V),
+        G.chromaticNumber = ⊤ →
+          ∃ (H : G.Subgraph), H.coe.chromaticNumber = ⊤ ∧ avoidsOddCyclesOfLength H.coe r := by
+  sorry
+
+end Erdos740