Covering $\mathbb R^{n+1}$ by graphs of $n$-ary functions and long linear orderings of Turing degrees

A point $(x_0,\dots,x_n)\in X^{n+1}$ is covered by a function $f:X^n\to X$ iff there is a permutation $\sigma$ of $n+1$ such that $x_{\sigma(0)}=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$.

By a theorem of Kuratowski, for every infinite cardinal $\kappa$ exactly $\kappa$ $n$-ary functions are needed to cover all of $(\kappa^{+n})^{n+1}$. We show that for arbitrarily large uncountable $\kappa$ it is consistent that the size of the continuum is $\kappa^{+n}$ and $\mathbb R^{n+1}$ is covered by $\kappa$ $n$-ary continuous functions.

We study other cardinal invariants of the $\sigma$-ideal on $\mathbb R^{n+1}$generated by continuous $n$-ary functions and finally relate the question of how many continuous functions are necessary to cover $\mathbb R^2$ to the least size of a set of parameters such that the Turing degrees relative to this set of parameters are linearly ordered.