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Let D = (X, D) be a Borel directed graph on a standard Borel space X and let χB(D) be its Borel chromatic number. If F0,. .. , Fn−1 : X → X are Borel functions, let DF 0 ,...,F n−1 be the directed graph that they generate. It is an open problem if χB(DF 0 ,...,F n−1) ∈ {1,. .. , 2n + 1, ℵ0}. This was verified for commuting functions with no fixed points. We show here that for commuting functions with the properties that χB(DF 0 ,...,F n−1) < ℵ0 and that there is a path from each x ∈ X to a fixed point of some Fj, there exists an increasing filtration {Xm}m<ω with X = m<ω Xm such that χB(DF 0 ,...,F n−1 Xm) ≤ 2n for each m. We also prove that if n = 2 in the previous case, then χB(DF 0 ,F 1) ≤ 4. It follows that the approximate measure chromatic number χ ap M (D) does not exceed 2n + 1 when the functions commute.

If X is a set, E is an equivalence relation on X, and n ∈ ω, then define [X] n E = {(x 0 , ..., x n−1 ) ∈ n X : (∀i, j)(i = j ⇒ ¬(x i E x j ))}. For n ∈ ω, a set X has the n-Jónsson property if and only if for every function f :A set X has the Jónsson property if and only for every function f :Let n ∈ ω, X be a Polish space, and E be an equivalence relation on X. E has the n-Mycielski property if and only if for all comeager C ⊆ n X, there is some ∆The following equivalence relations will be considered: E 0 is defined on ω 2 by x E 0 y if and only if (∃nwhere △ denotes the symmetric difference. E 3 is defined on ω ( ω 2) by x E 3 y if and only if (∀n)(x(n) E 0 y(n)).Holshouser and Jackson have shown that R is Jónsson under AD. It will be shown that E 0 does not have the 3-Mycielski property and that E 1 , E 2 , and E 3 do not have the 2-Mycielski property. Under ZF + AD, ω 2/E 0 does not have the 3-Jónsson property.