The definable (p,q)-theorem for dense pairs of certain geometric structures

Rakotonarivo, Tsinjo Odilon (2021-12)

Thesis (PhD)--Stellenbosch University, 2021.

Thesis

ENGLISH ABSTRACT: The definable (p, q)-conjecture is a model-theoretic version of a (p, q)- theorem in combinatorics, which was expressed in the form of a question by A. Chernikov and P. Simon in 2015. Researchers have proved that the property holds for certain classes of structures. Based on those existing results, the main objective of the present thesis is to show that the definable (p, q)-conjecture holds for a dense pair of geometric distal structures that satisfies the following condition: algebraic closure and definable closure are the same in sense of the original geometric structure. Independently, we also explore a different approach to prove that under some conditions, the definable (p, q)-conjecture holds in certain cases for dense pairs of real closed ordered fields.

AFRIKAANSE OPSOMMING: Die definieerbare (p, q)-vermoede is ’n model-teoretiese weergawe van ’n (p, q)-stelling in kombinatorika, wat in 2015 deur A. Chernikov en P. Simon in die vorm van ’n vraag uitgedruk is. Navorsers het bewys dat die eienskap vir seker klasse van strukture geldig is. Op grond van daardie bestaande resultate is die hoofdoel van hierdie tesis om aan te toon dat die definieerbare (p, q)-vermoede geldig is vir ’n digte paar meetkundige distale strukture wat die volgende voorwaarde bevredig: algebraïese afsluiting en definieerbare afsluiting is dieselfde in die sin van die oorspronklike meetkundige struktuur. Onafhanklik hiervan ondersoek ons ’n ander benadering om te bewys dat, onder seker voorwaardes, die definieerbare (p, q)-vermoede geldig is vir seker gevalle vir digte pare van reële geslote geordende liggame.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/123797
This item appears in the following collections: