# Largest possible families of a set of n elements

Let $$\mathcal F$$ be a family of subsets of a set consisting of $$n$$ elements so that no element of $$\mathcal F$$ is a subset of another element. Prove that $$\mathcal F$$ can have at most $$\binom{n}{\Big\lfloor\frac{n}{2}\Big\rfloor}.$$

Are these partitions? I suspected I could use induction, but it isn't really clear.

Award task, Socialist Republic Croatia, Yugoslavia 1973

• Sure, I'll do that. – Praskovya2.718281828 Nov 10 '19 at 20:22
• @ViktorGlombik I didn't want to write anything on my own because I translated it literally not to write anything wrong since I don't know if I have covered enough theory to conclude. – Praskovya2.718281828 Nov 10 '19 at 20:26
• I mean, this is obvious, but I followed the author... – Praskovya2.718281828 Nov 10 '19 at 20:28
• @ViktorGlombik I don't at all see how this would be equivalent to the sets in $\mathcal{F}$ being mutually disjoint. – Morgan Rodgers Nov 10 '19 at 20:45
• Oh, now I see how much I still have to learn. – Praskovya2.718281828 Nov 10 '19 at 20:46

## 1 Answer

This is just Sperner theorem.