Traces of hypergraphs

Let $\text{Tr}(n,m,k)$ denote the largest number of distinct projections onto $k$ coordinates guaranteed in any family of $m$ binary vectors of length $n$. The classical Sauer-Perles-Shelah Lemma implies that $\text{Tr}(n, n^r, k) = 2^k$ for $k \le r$. While determining $\text{Tr}(n,n^r,k)$ precisely for general $k$ seems hopeless even for constant $r$, estimating it, and more generally estimating the function $\text{Tr}(n,m,k)$ for all range of the parameters, remains a widely open problem with connections to important questions in computer science and combinatorics. Here we essentially resolve this problem when $k$ is linear and $m=n^r$ where $r$ is constant, proving that, for any constant $\alpha>0$, $\text{Tr}(n,n^r,\alpha n) = \tilde\Theta(n^C)$ with $C=C(r,\alpha)=\frac{r+1-\log(1+\alpha)}{2-\log(1+\alpha)}$. For the proof we establish a "sparse" version of another classical result, the Kruskal-Katona Theorem, which gives a stronger guarantee when the hypergraph does not induce dense sub-hypergraphs. Furthermore, we prove that the parameters in our sparse Kruskal-Katona theorem are essentially best possible. Finally, we mention two simple applications which may be of independent interest.