# A Combinatorial Structural Problem

Definition 1 ($MIF(k)$): A maximal intersecting family of k-sets [in short $MIF(k)$] is uniform intersecting family of k-sets such that if a k-set is not a member of family then there is at least a member in the family which has no common member with that k-set.

Definition 2 ($ISP(k)$): An intersecting family of k-sets say $(P, \mathbb B, \in)$ such that for each $B \in \mathbb B$ there exists a (k-1)-set $A^{'}$ such that $B$ and $A^{'}$ has no common element and $A^{'}$ intersects all the blocks except $B$.

Is it true that any $MIF(k)$ contains an $ISP(k)$ with the same point set $P$?

-
What's a $k$-set? Why does $P$ not occur in the explanation of $(P,\mathbb B,\in)$? How can an ISP and an MIF have the same points set $P$ if no point set is mentioned in the definition of an MIF? – joriki May 16 '12 at 12:11
@ to joriki: k-set is a set consisting of k elements. There is no need to express point set of any $MIF(k)$ it is intrinsic just union of the members of the family. – users31526 May 16 '12 at 12:38

This isn't true, simply because sometimes $P$ admits no $ISP(k)$ set! Namely, take $|P| < 2k-1$.
Even more explictly, take $|P| = k$, i.e. your $MIF(k)$ is just a single set $\{1, \ldots, k\}$.