Department of Mathematical Sciences

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Browsing Department of Mathematical Sciences by Subject "(p,q)-theorem"

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    The definable (p,q)-theorem for dense pairs of certain geometric structures
    (Stellenbosch : Stellenbosch University, 2021-12) Rakotonarivo, Tsinjo Odilon; Boxall, Gareth John; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.
    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.