Reducts of the random bipartite graph

TL;DR

This paper classifies the reducts of the random bipartite graph that preserve sides by analyzing permutation groups, using combinatorial theorems and finite submodel properties.

Contribution

It provides a classification of closed permutation subgroups containing automorphisms and anti-automorphisms of the bipartite graph, extending understanding of its symmetry structure.

Findings

Classified all reducts preserving sides of the random bipartite graph.

Identified the structure of permutation groups containing automorphisms and anti-automorphisms.

Utilized combinatorial theorems and finite submodel properties in the analysis.

Abstract

Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We classify the closed permutation subgroups containing the group $Aut(\Gamma)^*$, where $Aut(\Gamma)^*$ is the group of all isomorphisms and anti-isomorphisms of $\Gamma$ preserving the two sides. Our results rely on a combinatorial theorem of Ne\v{s}et\v{r}il-R\"{o}dl and a strong finite submodel property for $\Gamma$.