Here is a little problem for which I have no clue, and I don't even know if it is difficult.
Does there exist a measurable (!) function $\psi:[0,1]^2\mapsto [0,1]$ such that if $(X_i)_i$ is a sequence of iid uniform variables on $[0,1]$, then the $\psi(X_i,X_j), i< j$ are indepdendant (and of course identically distributed) variables?
The setting of the problem ($[0,1]$, uniform law, ...) can be changed, the only requirement is that the support of the $X_i$ and the arrival space have at least two values.
For info, the examples 1) $X_i$ are Bernoulli and $\psi(x,y)=x y$ and 2) $X_i$ are uniform and $\psi(x,y)$ is the congruence of $x+y$ modulo $1$ do not work.
Any remark is welcome too...