Reducts of the Generalized Random Bipartite Graph

TL;DR

This paper studies the reducts of a randomly colored bipartite graph with two vertex sets, classifying certain symmetry groups and relying on combinatorial and model-theoretic properties.

Contribution

It classifies the reducts of the generalized random bipartite graph that preserve vertex sets and describes the associated closed permutation groups.

Findings

Classification of reducts preserving Rl and Rr

Identification of closed permutation subgroups containing Aut(Gamma)

Application of Nesetril-Rodl theorem and finite submodel property

Abstract

Let \Gamma be the generalized random bipartite graph that has two sides Rl and Rr with edges for every pair of vertices between R1 and Rr but no edges within each side, where all the edges are randomly colored by three colors P1; P2; P3. In this paper, we investigate the reducts of \Gamma that preserve Rl and Rr, and classify the closed permutation subgroups in Sym(Rl)\timesSym(Rr) containing the group Aut(\Gamma). Our results rely on a combinatorial theorem of Nesetril-Rodl and the strong finite submodel property of the random bipartite graph.