Let $\|\,\,\,\|$ denote the Euclidean norm on $\mathbb R^2$. Let $\Lambda$ be a sublattice of $\mathbb Z^2$ and $m < M$ be positive real numbers. We say that a point $(x,y)$ in $\mathbb Z^2$ is primitive if $\gcd(x,y)=1$.
Define $\mathcal L_\Lambda(m,M)$ as a subcollection of primitive points $(x,y)$ in $\Lambda$ such that $m \leq \|(x,y)\|\leq M$, which is maximal with respect to the following property:
For any three distinct primitive points $(x_1,y_1),(x_2,y_2), (x_3,y_3) \in \mathcal L_\Lambda(m,M)$ there are no distinct $i,j,k\in \{1,2,3\}$ and no $\alpha,\beta \in \mathbb Z$ such that $(x_i,y_i) =\alpha (x_j,y_j)+\beta (x_k,y_k)$.
I am interested in obtaining "good" upper/lower bounds on the cardinality of $\mathcal L_\Lambda(m,M)$ in terms of $\Lambda, m, M$. This seems like a very natural question to ask, so I wanted to ask whether this problem was studied at all. Any suggestions on how to approach it would also be much appreciated.