2 problems

Let $D\subseteq [N]$ be $t$-intersective if every positive-density set $A\subseteq [N]$ contains a $(t+1)$-term arithmetic progression whose common difference lies in $D$. For the…

For every positive integer $k$ and every $\delta>0$, there exists a positive integer $n$ such that every subset $A\subset\{1,2,\dots,n\}$ of cardinality at least $\delta n$ contain…