Higher-arity distality and forking triviality

TL;DR

This paper proves that higher-arity triviality collapses to triviality in simple theories, and explores implications for $k$-distality, showing it is not a strict dividing line and is not preserved under reducts.

Contribution

It establishes a collapse result for $k$-triviality in simple theories and analyzes the implications for $k$-distality as a dividing line in model theory.

Findings

$k$-triviality collapses to triviality among simple theories.

No stable theory can be strictly $k$-distal for $k \\geq 3$.

Examples show $k$-distality is not preserved under reducts.

Abstract

Answering a question of Goode, we show that $k$-triviality collapses to (1-)triviality among simple theories. In particular, every stable theory with quantifier elimination in a relational language of bounded arity is trivial. We use our collapse result, along with other facts about $k$-triviality and $k$-total triviality, to generate examples of (strongly) $k$-distal theories. The collapse result immediately implies that no stable theory can be strictly $k$-distal for some $k\geq 3$, partially answering a question of Walker. Moreover, all known examples of non-distal (strongly) $k$-distal theories are $k$-ary, rendering (strong) $k$-distality moot as a $(k+1)$-ary dividing line; we give four classes of examples that are not $k$-ary. We also show that just as distality is not preserved under taking reducts, neither is (strong) $k$-distality.