Borel Chromatic Numbers

Ultraproducts of measure preserving actions and graph combinatorics

“…Let d = |S ±1 | be the degree of Cay( , S). First note that by Kechris et al [KST,4.8], χ µ (S, a) ≤ d + 1 (in fact, this holds even for Borel instead of measurable colorings). A compactness argument using Brooks' theorem also shows that χ (S, a) ≤ d, where χ (S, a) is the chromatic number of G(S, a).…”

“…The above argument can be simplified by using [KST,4.6]. Consider the graph G n on X n whose edges consist of all distinct x, y such that y = (γ ±1 ) α n · x.…”

Finite versus infinite: An insufficient shift

“…We show that this graph is not a minimum among the graphs of the form G f defined on some Polish space X, where two distinct points are adjacent if one can be obtained from the other by a given Borel function f : X Ñ X. This answers the primary outstanding question from [KST99].A directed graph is a pair G " pX, Rq where R is an irreflexive binary relation on X. A homomorphism from G " pX, Rq to G 1 " pX 1 , R 1 q is a map h : X Ñ X 1 such that px, yq P R implies phpxq, hpyqq P R 1 for all x, y P X.…”

“…We show that this graph is not a minimum among the graphs of the form G f defined on some Polish space X, where two distinct points are adjacent if one can be obtained from the other by a given Borel function f : X Ñ X. This answers the primary outstanding question from [KST99].…”

“…Kechris, Solecki and Todorčević [KST99,Problem 8.1] (see also [DPT15, Section 3] and [Mil08]) asked whether the following is true: If X is a Polish space and f : X Ñ X is a Borel function, then exactly one of the following holds:…”