# Cardinality Of Set Where No Sum Of Elements Equals Some $k$

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.)

• Ignoring rounding, there are about $\frac k2$ pairs which sum to $k$. You can't choose both members from any pair, so... – lulu Oct 6 at 21:22
• Can you improve your problema statement, please? "$\forall A\subset S: ~ \sum_{i\in A} i \neq k$ ...." or " $\sum_{i\in S} i\neq k$" ? – Alexandre Frias Oct 6 at 22:01
• For some set of numbers less than $k$, no subset of that set sums to $k$. Is that clear? – Lieutenant Zipp Oct 6 at 23:26