Consequences of Dependent Dividing on Burden
Yuki Takahashi
TL;DR
The paper studies dependent dividing in $NTP_2$ theories and shows that the burden agrees with the $NIP$-$dp$-rank witnessed by $NIP$ formulas when dependent dividing holds. It then proves sub-additivity of the burden, first in the $NIP$-$dp$-minimal case and then for $NIP$-$dp$-finite theories, using adaptations of dp-rank techniques and finite-pattern arguments. An alternative proof via a reduct to a complete $NIP$ theory is provided. Finally, the work connects the burden to $VC^\ast$-density, showing that under dependent dividing, the burden lower bounds correspond to dual VC density for $NIP$ formulas.
Abstract
If $T$ has dependent dividing, then the burden agrees with the dp-rank witnessed by NIP formulas. We use this observation to prove that if $T$ has dependent dividing, then the burden is sub-additive. We also state a connection between the burden and the dual VC density.