When considering something completely different yesterday I came across the following problem:
Let $X = \{ 1,2,3,...,n\}$. What is the maximal number of subsets of $X$ of order $m$ one can choose such that any two chosen subsets have at most $k$ elements in common?
If we call this number $F(n,m,k)$, it is quite easy to calculate $F$ for some low values of $n$,$m$ and $k$, such as:
- $F(3,3,1) = 1$
- $F(4,3,1) = 1$
- $F(5,3,1) = 2$
- $F(6,3,1) = 3$
- $F(7,3,1) = 6$
and also $F(n,n,k)=1$, but I cannot seem to find anything general at all. Am I missing something trivial?