Assuming $r=\lambda^2$, here is a stand-alone example with $r=4$: the blocks are 1234, 12, 13, 14, 23, 24, 34. Note that the blocks 12 and 34 are disjoint.

Assuming $r=\lambda^2$, here is a stand-alone example with $r=4$: the blocks are 1234, 12, 13, 14, 23, 24, 34. Note that the blocks 12 and 34 are disjoint.

EDIT: One can extend this example to 12345, 125, 13, 14, 23, 24, 345, 5

and then to 123456, 125, 136, 14, 23, 246, 345, 56

and then to 123456, 1257, 1367, 14, 23, 2467, 3457, 56.