  A $k$-uniform hypergraph with $n$ vertices is an $(n,k,\ell)$-omitting system
if it does not contain two edges whose intersection has size exactly $\ell$. If
in addition it does not contain two edges whose intersection has size greater
than $\ell$, then it is an $(n,k,\ell)$-system. R\"{o}dl and
\v{S}i\v{n}ajov\'{a} proved a lower bound for the independence number of
$(n,k,\ell)$-systems that is sharp in order of magnitude for fixed $2 \le \ell
\le k-1$. We consider the same question for the larger class of
$(n,k,\ell)$-omitting systems.
  For $k\le 2\ell+1$, we believe that the behavior is similar to the case of
$(n,k,\ell)$-systems and prove a nontrivial lower bound for the first open case
$\ell=k-2$. For $k>2\ell+1$ we give new lower and upper bounds which show that
the minimum independence number of $(n,k,\ell)$-omitting systems has a very
different behavior than for $(n,k,\ell)$-systems. Our lower bound for
$\ell=k-2$ uses some adaptations of the random greedy independent set
algorithm, and our upper bounds (constructions) for $k> 2\ell+1$ are obtained
from some pseudorandom graphs.
  We also prove some related results where we forbid more than two edges with a
prescribed common intersection size and this leads to some applications in
Ramsey theory. For example, we obtain good bounds for the Ramsey number
$r_{k}(F^{k},t)$, where $F^{k}$ is the $k$-uniform Fan. Here the behavior is
quite different than the case $k=2$ which reduces to the classical graph Ramsey
number $r(3,t)$.
