  We study $r$-differential posets, a class of combinatorial objects introduced
in 1988 by the first author, which gathers together a number of remarkable
combinatorial and algebraic properties, and generalizes important examples of
ranked posets, including the Young lattice. We first provide a simple bijection
relating differential posets to a certain class of hypergraphs, including all
finite projective planes, which are shown to be naturally embedded in the
initial ranks of some differential poset. As a byproduct, we prove the
existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic
in any two consecutive ranks but having the same rank function. We also show
that the Interval Property, conjectured by the second author and collaborators
for several sequences of interest in combinatorics and combinatorial algebra,
in general fails for differential posets. In the second part, we prove that the
rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial
growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the
Hardy-Ramanujan asymptotic formula that holds in the special case of Young's
lattice. We conclude by posing several open questions.

Combinatorics;Commutative Algebra