For a large enough $n$, and a parameter $ m $ I'm looking for a subset of the prime numbers in the range $[n,2n]$ with a unique structure. I am looking for a prime $p$ and a set of $m$ deltas: $\Delta_1,\Delta_2,\ldots,\Delta_m$ such that the following set consists only of prime numbers in the range $[n,2n]$: $$\{{ p+\sum_{i=1}^{m} a_i \cdot \Delta_i} \mid a_i \in \{0,1\} \}$$ For example, for $n=10$ and $m=2$ there is such a set with parameters: $$p=11,\quad \Delta_1 = 2,\quad \Delta_2 = 6$$ And the set is: $$\{11,13,17,19\} $$ The question is how large can $m$ be asymptotically? Note that this is a generalization of arithmetic progression of primes. I've managed to prove that $m$ could be $\frac{\log\log(n)}{2}$ though I'm sure my bound is far from tight.

For a large enough $n$, and a parameter $ m $ I'm looking for a subset of the prime numbers in the range $[n,2n]$ with a unique structure. I am looking for a prime $p$ and a set of $m$ positive (not necessary different) integers: $\Delta_1,\Delta_2,\ldots,\Delta_m$ such that the following set consists only of prime numbers in the range $[n,2n]$: $$\{{ p+\sum_{i=1}^{m} a_i \cdot \Delta_i} \mid a_i \in \{0,1\} \}$$ For example, for $n=10$ and $m=2$ there is such a set with parameters: $$p=11,\quad \Delta_1 = 2,\quad \Delta_2 = 6$$ And the set is: $$\{11,13,17,19\} $$ The question is how large can $m$ be asymptotically? Note that an arithmetic progression of $m+1$ primes is a cube with all $m$ deltas having the same value. I've managed to prove that $m$ could be $\frac{\log\log(n)}{2}$ though I'm sure my bound is far from tight.