Freiman conjectured that if S is a finite subset of a torsion-free group G with \(k\ge 3\) elements and \(|S^{2}|\le 3k-4,\) then S is a subset of a small geometric progression of length at most \(2k-3\) . In 2014, Freiman et al. settled this conjecture when S is a finite subset of an ordered group. In this article, we focus on right-ordered groups and prove Freiman’s conjecture when S is a finite subset of a right-ordered group under certain restrictions on the set S. Further, we consider the right-ordered Baumslag-Solitar group \(\textrm{BS}(1,q) =\langle a,b: ab = b^qa\rangle \) , where q is an integer. We answer Freiman’s conjecture for \(\textrm{BS}(1,q)\) , \(q\ne -1\) by proving that if S is a finite subset of \(\textrm{BS}(1,q)\) , \(q\ne -1\) with the identity element as its minimum and \(|S^2|\le 3|S|-4\) , then the subgroup generated by the set S is an abelian subgroup of \(\textrm{BS}(1,q)\) .
