The measure half of the $2n+1$ problem

RESUMO

Let $F_0,\dots,F_{n-1}$ be Borel functions on a standard Borel space $X$ and let $G_F$ be the graph they generate. We prove that for every Borel probability measure $\mu$ on $X$ there are a forward-invariant $\mu$-conull Borel set $A \subseteq X$ and a proper Borel $(2n+1)$-coloring of the restriction of $G_F$ to $A$; consequently $\chi_M(G_F) \le 2n+1$ in the total sense of Kechris and Marks. This answers positively, for every $n$ and with the optimal constant, the measure half of Problem 5.14 of the survey "Descriptive graph combinatorics" of Kechris and Marks -- the measurable version of a question raised by Kechris, Solecki, and Todorcevic (Adv. Math. 141, 1999). No invariance, no local finiteness, and no finiteness of the Borel chromatic number are assumed. The argument is elementary and specific to measure; the corresponding Borel problem and the Baire-measurable half remain open.