Does there exist an uncountable set of subsets of the positive integers such that any two distinct subsets have finite intersection?
Consider first the set Q of rationals. For each positive real in the interval (0, 1), take a sequence of rationals converging to it. This gives an uncountable collection of subsets of Q. Clearly two collections have at most finitely many points in common. To get the corresponding subsets of the postive integers simply take a bijection between the positive integers and the rationals.
50th Putnam 1989
© John Scholes
1 Jan 2001