COMBINATORIAL BOUNDS IN DISTAL STRUCTURES
Aaron Anderson · Journal of Symbolic Logic · 2023
Abstract We provide polynomial upper bounds for the minimal sizes of distal cell decompositions in several kinds of distal structures, particularly weakly o -minimal and P -minimal structures. The bound in general weakly o -minimal structures generalizes the vertical cell decomposition for semialgebraic sets, and the bounds for vector spaces in both o -minimal and p -adic cases are tight. We apply these bounds to Zarankiewicz’s problem in distal structures.