A set $T\subset \mathbb{N}$ is called thick if it contains arbitrarily long intervals. A set $S \subset \mathbb{N}$ is called syndetic if it has bounded gaps.
I am trying to prove that a set $T$ is thick if and only if $S\cap T \neq \emptyset$ for any synetic set $S.$
Forward direction was fairly easy: If $T\subset \mathbb{N}$ is thick then for any syndetic set $S$ there exists $M\in \mathbb{N}$ such that $S$ has nontrivial intersection with every interval of length $M$ in $\mathbb{N}.$ Some interval of length $M$ is guaranteed to be contained in $T$ thus intersecting $S.$
I am struggling with the other side. I couldn't proceed further than rewriting definitions.
Now, the condition that $S \cap T \neq\emptyset$ for each syndetic set tells you that, in particular, the complement of $T,$ call it $T^c,$ is not syndetic. Therefore it contains arbitrarily large gaps. But this means that $T$ must have arbitrarily long intervals!