Erdős problem 198 · problems.science

The answer is no; Erdős and Graham report this was proved by Baumgartner, presumably referring to the paper [Ba75], which does not state this exactly, but the following simple construction is implicit in [Ba75].

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198.lean

Retained formal statement2 of 4

Let $V$ be a vector space over the rationals and let $k$ be a fixed positive integer. Then there is a set $X_k \subseteq V$ such that $X_k$ meets every infinite arithmetic progression in $V$ but $X_k$ intersects every $k$ -element arithmetic progression in at most two points.

At the end of [Ba75] the author claims that by "slightly modifying the method of [his proof]", one can prove this.

FormalConjectures/ErdosProblems/198.leanErdos198.baumgartner_strong2 linesExact file

1∀ (V : Type u_1) [inst : AddCommGroup V] [Module ℚ V] (k : ℕ),2  ∃ X, (∀ (Y : Set V), Y.IsAPOfLength ⊤ → (X ∩ Y).Nonempty) ∧ ∀ (Y : Set V), Y.IsAPOfLength ↑k → (X ∩ Y).ncard ≤ 2

SolvedStatement only, no proof

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