Covering ℝ ⁿ⁺¹ by graphs of n-ary functions and long linear orderings of turing degrees

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 (κ ⁺ⁿ) ⁿ⁺¹. We show that for arbitrarily large uncountable κ it is consistent that the size of the continuum is κ ⁺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. Continuous function, n-space, forcing extension, covering number, Turing degree.