Suppose $m$ is a positive integer.


I am looking for finite sets with group actions such that the action is transitive on the set of $m$-element subsets, but NOT transitive on the set of $(m+1)$-element subsets.


An example for $m=2$ is a projective space over a finite field.



Not very transitive actions
According to Theorem 4.11 of Peter Cameron's book `Permutation Groups' it follows from the classification of finite simple groups that the only finite 6-transitive groups are (some of the) symmetric and alternating groups in their natural actions, and the only finite 4-transitive groups are symmetric, alternating and the Mathieu groups $M\_{11}$, $M\_{12}$, $M\_{23}$ and $M\_{24}$. Thus there will not be very many examples of what you are looking for when $m\geq 4$.


