Countably Colorful Hyperplane Transversal

Let \ Fₙ ∈ N be an infinite sequence of families of compact connected sets in Rᵈ. An infinite sequence of compact connected sets \ Bₙ ∈ N is called heterochromatic sequence from \ Fₙ ∈ N if there exists an infinite sequence \ iₙ ∈ N of natural numbers satisfying the following two properties: (a) ₙ∈ N is a monotonically increasing sequence, and (b) for all n ∈ N, we have Bₙ ∈ Fiₙ. We show that if every heterochromatic sequence from \ Fₙ ∈ N contains $d+1$ sets that can be pierced by a single hyperplane then there exists a finite collection H of hyperplanes from Rᵈ that pierces all but finitely many families from \ Fₙ ∈ N. As a direct consequence of our result, we get that if every countable subcollection from an infinite family F of compact connected sets in Rᵈ contains $d+1$ sets that can be pierced by a single hyperplane then F can be pierced by finitely many hyperplanes. To establish the optimality of our result we show that, for all d ∈ N, there exists an infinite sequence \ Fₙ ∈ N of families of compact connected sets satisfying the following two conditions: (1) for all n ∈ N, Fₙ is not pierceable by finitely many hyperplanes, and (2) for any m ∈ N and every sequence ₙ\ₙ₌ₘ∞ of compact connected sets in Rᵈ, where Bᵢᵢ for all i ≥ m, there exists a hyperplane in Rᵈ that pierces at least $d+1$ sets in the sequence.