Almost-everywhere totality of the induced order

Establish whether there exists a full-measure subset \(A\subseteq Z(\varnothing)\) on which the relation \(\prec\), defined by \(x\prec y\) when \(\{x,y\}\cap Z(\{x,y\})=\{x\}\), is a genuine total order.

Background

In the proof of Theorem 1.3, the authors define an asymmetric relation $\prec$ on the points of $Z(\varnothing)$ according to which point is selected by the stopping set when presented with a two-point configuration. Claim 5 proves transitivity almost everywhere, so the relation is an almost total order in the measure-theoretic sense.

The construction of the exploration attaining the stopping set does not require the authors to resolve whether this relation can be made into an actual total order on one fixed full-measure subset. The paper explicitly identifies that measurability and order-theoretic question as unresolved.