Countably Colorful Hyperplane Transversal

Let \ F₍\₍ ₍ be an infinite sequence of families of compact connected sets in R^d. An infinite sequence of compact connected sets \ B₍ \₍ ₍ is called heterochromatic sequence from \ F₍\₍ ₍ if there exists an infinite sequence \ i₍ \₍ ₍ of natural numbers satisfying the following two properties: (a) \i₍\₍ ₍ is a monotonically increasing sequence, and (b) for all n N, we have B₍ F₈䂸. We show that if every heterochromatic sequence from \ F₍\₍ ₍ contains d+1 sets that can be pierced by a single hyperplane then there exists a finite collection H of hyperplanes from R^d that pierces all but finitely many families from \ F₍\₍ ₍. As a direct consequence of our result, we get that if every countable subcollection from an infinite family F of compact connected sets in R^d 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 \ F₍\₍ ₍ of families of compact connected sets satisfying the following two conditions: (1) for all n N, F₍ is not pierceable by finitely many hyperplanes, and (2) for any m N and every sequence \Bₙ\₍=₌^ 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.