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

Abstract

A point (x 0 ,...,x n ) ∈ X n+1 is covered by a function f: X n → X iff there is a permutation σ of n + 1 such that x σ(0) = f(x σ(1) ,...,x σ(n) ). By a theorem of Kuratowski, for every infinite cardinal κ exactly κ n-ary functions are needed to cover all of (K +n ) n+1 . We show that for arbitrarily large uncountable κ it is consistent that the size of the continuum is C +n and R n+1 is covered by κ n-ary continuous functions. We study other cardinal invariants of the σ-ideal on R n+1 generated by continuous n-ary functions and finally relate the question of how many continuous functions are necessary to cover 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.