Let S = {1, .. ,n}.

Let H = (S, E) be the m-uniform hypergraph with r edges.
Let F(H, k) = #{B | |B| = k, $\exists R \in E, B \cap R = \emptyset$ } -
a number of k-subsets, that doesn't intersect with edges of H.

I am interested in following two problems:

Let r, m, k be fixed parameters.

- For which hypergraph H, F(H, k) be minimal?
- For which H, F(H, k) be maximal?

I have a conjecture for the first problem, but can't prove it.

Introduce a total order $\preceq$ on m-element subsets of S: $a \preceq b$ iff $\exists s$, such that $s \in a, s \notin b$ and $\forall i < s, i \in a \cap b \mbox{ or } i \notin a \cup b$.

Example: 3-subsets from 5 element set in ascending order according to $\preceq$.

11100

11010

11001

10110

10101

10011

01110

01101

01011

01111

ConjectureFor all k minimal value of F(H, k) achieved on hypergraph, whose edges are r first sets from ${S \choose m}$ according to $\preceq$ order.