  An $r$-uniform hypergraph is called $t$-cancellative if for any $t+2$
distinct edges $A_1,\ldots,A_t,B,C$, it holds that $(\cup_{i=1}^t A_i)\cup
B\neq (\cup_{i=1}^t A_i)\cup C$. It is called $t$-union-free if for any two
distinct subsets $\mathcal{A},\mathcal{B}$, each consisting of at most $t$
edges, it holds that $\cup_{A\in\mathcal{A}} A\neq \cup_{B\in\mathcal{B}} B$.
Let $C_t(n,r)$ (resp. $U_t(n,r)$) denote the maximum number of edges of a
$t$-cancellative (resp. $t$-union-free) $r$-uniform hypergraph on $n$ vertices.
Among other results, we show that for fixed $r\ge 3,t\ge 3$ and
$n\rightarrow\infty$
$$\Omega(n^{\lfloor\frac{2r}{t+2}\rfloor+\frac{2r\pmod{t+2}}{t+1}})=C_t(n,r)=O(n^{\lceil\frac{r}{\lfloor
t/2\rfloor+1}\rceil})\text{ and }
\Omega(n^{\frac{r}{t-1}})=U_t(n,r)=O(n^{\lceil\frac{r}{t-1}\rceil}),$$ thereby
significantly narrowing the gap between the previously known lower and upper
bounds. In particular, we determine the Tur\'an exponent of $C_t(n,r)$ when
$2\mid t \text{ and } (t/2+1)\mid r$, and of $U_t(n,r)$ when $(t-1)\mid r$.
  The main tool used in proving the two lower bounds is a novel connection
between these problems and sparse hypergraphs.
