Problem #56: Let $N\geq p_k$ where $p_k$ is the $k$th prime

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Progress

This was disproved for $k=212$ by Ahlswede and Khachatrian [AhKh94], who suggest that their methods can disprove this for arbitrarily large $k$.

Erdős later asked ([Er92b] and [Er95]) if the conjecture remains true provided $N\geq (1+o(1))p_k^2$ (or, in a weaker form, whether it is true for $N$ sufficiently large depending on $k$).

Ahlswede and Khachatrian [AhKh95] proved this latter claim: in other words, for any fixed $k$, if $N$ is sufficiently large depending on $k$ then the largest such set is the set of all multiples of the first $k$ primes.

See also [534].

This is discussed in problem B26 of Guy's collection [Gu04].