What is the greatest cardinality of a set $S$ of positive integers strictly less than $k$ such that no subset of $S$ sums to $k$, in terms of $k$?
I worked this out for small values of $k$, and realized that a lower bound for the cardinality of $S$ is $\lfloor \frac{k}{2} \rfloor$, as one can select $S$ to be the set of integers $n$ such that $\frac{k}{2} \le n \lt k$ (and then any value of $n \lt \frac{k}{2}$ would have a corresponding element of $S$ equal to $k-n$ for which the sum is $k$), but I want to know if there is a better bound, and if so, how to show that it holds. (For the record, I don’t think that one exists, but I don’t know how to prove that.)