# Combinatorial Bounds in Distal Structures

@inproceedings{Anderson2021CombinatorialBI, title={Combinatorial Bounds in Distal Structures}, author={Aaron Anderson}, year={2021} }

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 and sum-product bounds in distal structures.

## One Citation

### Semi-equational theories

- Mathematics
- 2022

. We introduce and study semi-equational and weakly semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a…

## References

SHOWING 1-10 OF 32 REFERENCES

### Computations of Vapnik–Chervonenkis Density in Various Model-Theoretic Structures

- Mathematics, Computer ScienceThe Bulletin of Symbolic Logic
- 2018

In the theory of infinite trees the authors establish an optimal bound on the VC-density function and show that superflat graphs are dp-minimal, following the results of Podewski and Ziegler.

### Cutting lemma and Zarankiewicz’s problem in distal structures

- MathematicsSelecta Mathematica
- 2020

A cutting lemma is established for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definite families of set on the plane in o-minimal expansions of fields, which generalizes the results in Fox et al.

### Regularity lemma for distal structures

- MathematicsJournal of the European Mathematical Society
- 2018

It is known that families of graphs with a semialgebraic edge relation of bounded complexity satisfy much stronger regularity properties than arbitrary graphs, and that they can be decomposed into…

### A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications

- Computer ScienceTheor. Comput. Sci.
- 1991

### Vapnik-Chervonenkis Density in Some Theories without the Independence Property, II

- MathematicsNotre Dame J. Formal Log.
- 2013

The problem of calculating Vapnik-Chervonenkis (VC) density is recast into one of counting types, and bounds (often optimal) on the VC density are calculated for some weakly o-minimal, weakly quasi-o- Minimal, and $P$-Minimal theories.

### One‐Dimensional p‐Adic Subanalytic Sets

- Mathematics
- 1999

In this paper we extend two theorems from [2] on p‐adic subanalytic sets, where p is a fixed prime number, Qp is the field of p‐adic numbers and Zp is the ring of p‐adic integers. One of these…

### Reducts of p-adically closed fields

- MathematicsArchive for Mathematical Logic
- 2014

In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets Mf={(x,y)∈K2∣|y|=|f(x)|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}…

### Incidence bounds in positive characteristic via valuations and distality

- Mathematics
- 2021

. We prove distality of quantiﬁer-free relations on valued ﬁelds with ﬁnite residue ﬁeld. By a result of Chernikov-Galvin-Starchenko, this yields Szemer´edi-Trotter-like incidence bounds for function…

### Distality in valued fields and related structures

- Mathematics
- 2020

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an…

### The polynomial method over varieties

- MathematicsInventiones mathematicae
- 2020

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number…