Oscillation stability for continuous monotone surjections

We prove that for every real ε > 0 there exists a positive integer t such that for every finite coloring of the nondecreasing surjections from [ 0 , 1 ] onto [ 0 , 1 ] there exist t many colors such that their ε -fattening contains a cube, i.e. a set of the form { f ∘ h : f nondecreasing surjection from [ 0 , 1 ] onto [ 0 , 1 ] } where h is a nondecreasing surjection from [ 0 , 1 ] onto [ 0 , 1 ] . We prove this as a consequence of a corresponding result about b ω and we determine the minimal integer t = t ( ε ) that works for a given ε > 0 .