nt.number theory - Can select many disjoint pairs with prescribed differences from Z_n?

Suppose we have a sequence $d_i<2n$ for $i=1,ldots,n$ and we want to select $n$ disjoint pairs from $Z_p$, $x_i,y_i$ such that $x_i-y_i=d_i mod p$. Then how big $p$ has to be compared to $n$ to do this? I am primary interested on an upper bound on $p$. Is it true that there is always a $ple (1+epsilon)2n+O(1)$?

My comments. It is trivial that $pge 2n$ because all the numbers $x_i,y_i$ must be different and $d_1=1, d_2=2$ shows that this is not always enough. I also guess that it helps if $p$ is a prime, maybe the smallest prime bigger than $2n$ works which would answer the question.