On the length of D(±1)$D(\pm 1)$‐tuples in imaginary quadratic rings

Let K$K$ be an imaginary quadratic field and OK$\mathcal {O}_K$ its ring of integers. Fix a rational integer ε$\varepsilon$. A set {a1,a2,⋯,am}⊂OK∖{0}$\lbrace a_1, a_2, \dots ,a_m\rbrace \subset \mathcal {O}_K\setminus \lbrace 0\rbrace$ is called a D(ε)$D(\varepsilon )$‐m$m$‐tuple in OK$\mathcal {O}_K$ if aiaj+ε=xij2$a_ia_j +\varepsilon = x_{ij}^2$, where xij∈OK$x_{ij} \in \mathcal {O}_K$ for all i,j$i,j$ such that 1⩽i24$m > 24$.