How many of the triangle inequality constraints of a discrete metric space are redundant?

Consider a discrete set of points $$X$$ and a distance function $$d : X \times X \to \mathbb{R}_+$$. $$d(\cdot,\cdot)$$ is said to be a metric over $$X$$ if the following three constraints are satisfied:

1. $$\forall x,y \in X,\ d(x,y) = 0 \Leftrightarrow x = y$$.
2. $$\forall x,y \in X,\ d (x,y) = d (y,x)$$ [symmetry].
3. $$\forall x,y,z \in X,\ d(x,y) + d(y,z) \ge d (x,z)$$ [triangle inequality].

A $$k$$-metric space is $$X$$ equipped with a distance function that always satisfies the first two constraints, and $$k$$ of the $$\binom{|X|}{2} (|\mathcal{X}| - 2)$$ non-trivial$$^*$$ triangle inequality constraints.

What is the maximum $$k$$ such that there exists a $$k$$-metric space that is not a metric space?

$$*$$ A set of triangle inequalities is non-trivial if each is composed of distinct $$x,y,z$$ and inequalities remain distinct upon all possible symmetry transformations.

• Where does $\binom {|X|}3$ come from? The roles of $x$ and $z$ can be swapped (by the symmetry axiom), but $y$ is in a different role. I would think that there are essentially $(|X|-2)\binom{|X|}2$ non-trivial triangle inequalities. Commented Feb 14, 2019 at 15:18
• Or was the idea to lump together the three inequalities involving the same triangle? Commented Feb 14, 2019 at 15:19
• @JyrkiLahtonen You are right. My intention was to look at all triangle inequalities composed of different points, that remain distinct from each other upon all possible symmetry transformations. Commented Feb 14, 2019 at 15:23

From what I understand, if we let $$\text{d}(x,y)=1$$ when $$x\neq y$$, except for the distance between $$a$$ to $$b$$ and $$a$$ to $$c$$ for some fixed $$a,b, \text{ and } c \in X$$; where we have $$\text{d}(a,b)=0.1$$ and $$\text{d}(a,c)=0.1$$, we will have $$X$$ as a $$[\binom{|X|}{2}(|X|-2)-1]$$-metric space.
• This looks correct to me. Except the value of $k$ as $\binom{|X|}{3} - 1$ (refer to updated question). Commented Feb 14, 2019 at 15:30