In \(n\)-dimensional Hamming space three points are on a line, if they satisfy the triangle inequality with equality. The paper introduces the following problem: How many different points can be found in the Hamming space so that no three of them are on a line (that is they are in general position)? This maximum value is \(A(n)\). The paper surveys the relation among \(A(n)\) and a number of well-known quantities from Hamming spaces and extremal set theory. Furthermore, the paper establishes inequalities for the exponential asymptotics \[ \alpha= \limsup_{n\to\infty} \textstyle{{1\over n}}\log A(n) \] showing that \(0,21\leq \alpha\leq 1/2\), improving some well-known older results.