  For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is
$E$-intersective if for every $A \subset E$ having positive upper relative
density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is
chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k
E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When
$E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic
intersectivity.
  In this article, we investigate to which extent known intersectivity results
hold in the relative setting when $E = \mathbb{P}$, the set of primes, or other
sparse subsets of $\mathbb{N}$. Among other things, we prove:
  -There exists an intersective set that is not $\mathbb{P}$-intersective.
  -However, every $\mathbb{P}$-intersective set is intersective.
  -There exists a chromatically $\mathbb{P}$-intersective set which is not
intersective (and therefore not $\mathbb{P}$-intersective).
  -The set of shifted Chen primes $\mathbb{P}_{\mathrm{Chen}} + 1$ is
$\mathbb{P}$-intersective (and therefore intersective).
