erdos_198

If $A \subseteq \mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression? Answer "yes" according to remark on page 23 of: - Erdös and Graham, "Old and new problems and results in combinatorial number theory", 1980. "Baumgartner also proved the conjecture of Erdös that if $A$ is a sequence of positive integers with all sums $a + a'$ distinct for $a, a' \in A$ then the complement of $A$ contains an infinite A.P." But this seems to be a misprint, since the opposite is true: There is a sequence of positive integers with all $a + a'$ distinct for $a, a' \in A$ such that the complement of $A$ contains no infinite A.P., i.e. there is a Sidon set $A$ which intersects all arithmetic progressions. So the answer should be "no". This can be seen, as pointed out by Thomas Bloom [erdosproblems.com/198](https://www.erdosproblems.com/198), by an elementary argument.

theorem erdos_198 : (∀ A : Set ℕ, IsSidon A → (∃ Y, IsAPOfLength Y ⊤ ∧ Y ⊆ Aᶜ)) ↔
    answer(False)

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