For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon D^+_R\to\mathbb{R}$ such that for any $n$ points $(x_1,y_1),\dots,(x_n,y_n)$ in $D^+_R$ the indicators $I_{(x_1,y_1]},\dots,I_{(x_n,y_n]}$ of the intervals $(x_1,y_1],\dots,(x_n,y_n]$ are linearly independent whenever the values $f(x_1,y_1),\dots,f(x_n,y_n)$ of the function $f$ at the points $(x_1,y_1),\dots,(x_n,y_n)$ are pairwise distinct.
In particular, the sets $F_{1,R}$ and $F_{2,R}$ coincide with the entire set of all Borel-measurable functions $f\colon D^+_R\to\mathbb{R}$, whereas $F_{3,R}$ is the set of all Borel-measurable functions $f\colon D^+_R\to\mathbb{R}$ such that, for any $a,b,c$ in $R$ satisfying the condition $a<b<c$, the values $f(a,b),f(b,c),f(a,c)$ are not pairwise distinct.
Let $F_{\infty,R}:=\bigcap_{n=1}^\infty F_{n,R}$.
Is there an explicit characterization of any of the sets $F_{n,R}$ with $n=3,4,\dots,\infty$ (the set $F_{\infty,\mathbb{R}}$ being of particular interest)?
It is clear that, for any $n=1,2,\dots,\infty$, any $f\in F_{n,R}$, and any Borel-measurable function $g\colon\mathbb{R}\to\mathbb{R}$, one has $g\circ f\in F_{n,R}$. So, writing $f\succeq h$ iff $h=g\circ f$, one may try to find a maximal (with respect to the order $\succeq$) function $f\in F_{n,R}$ or perhaps characterize a sigma-algebra $\Sigma$ over $D^+_R$ (if it exists) such that $f\in F_{n,R}$ iff $f$ is $\Sigma$-measurable.
This problem arises in a study of statistical estimators.
