Kruskal-Katona type Problem

The Kruskal Katona theorem was proved in the 1960s. In the theorem, we are given an integer $r$ and families of sets $\mathcal{A}\subset \mathbb{N}^{(r)}$ and $\mathcal{B}\subset\mathbb{N}^{(r-1)}$ such that for every $A\in\mathcal{A}$, every subset of $A$ of size $r-1$ is in $\mathcal{B}$. We are interested in finding the mimimum size of $b=|\mathcal{B}|$ given fixed values of $r$ and $a=|\mathcal{A}|$. The Kruskal Katona theorem states that this mimimum occurs when both $\mathcal{A}$ and $\mathcal{B}$ are initial segments of the colexicographic ordering. The Kruskal Katona theorem is very useful and has had many applications and generalisations. In this paper, we are interested in one particular generalisation, where instead of every subset of $A$ of size $r-1$ being in $\mathcal{B}$, we will instead ask that only $k$ of them are, where $k$ is some integer smaller than $r$. Note that setting $k=r$ is exactly the Kruskal Katona theorem. We will first find exact results for the cases where $0\leq k \leq 3$. For $k\geq 4$ we will not solve the question completely, however, we will find the exact result for infinitely many $a$. We will also provide a formula that is within some additive constant of the correct result for all $a$.