Department of Quantitative Management, University of Sout h Africa

Title: Department of Quantitative Management, University of Sout h Africa

Research Question: How do random binary strings, known as Kolmogorov-Chaitin (KC) strings, interact with countable homogeneous structures, specifically, the random graph R of Rado?

Methodology: The study uses Fra¨ ıss´ e's characterization of countable homogeneous structures and a recursive representation of the random graph R. It employs the concept of KC-strings, which are random partitions of the copies of a given structure within R.

Results: The main result is that for any KC-string ε, there exists a monochromatic β-organization Yε in R, which is a subset of the copies of β within R. This implies that KC-strings preserve the symmetric structure of R in two distinct ways: as a random partition and as a generator of a β-organization that is monochromatic under this partition.

Implications: This research provides a new perspective on the interaction between randomness and symmetry in countable homogeneous structures. It also offers a method to generate the Fra¨ ıss´ e limit of ranked diagrams using KC-strings.

Significance: The findings of this research have implications in various fields, including computer science, mathematics, and complex systems. It contributes to the understanding of randomness and symmetry in complex structures and may have applications in modeling and analyzing real-world systems.

Link to Article: https://arxiv.org/abs/0402034v1 Authors: arXiv ID: 0402034v1