Köhler theory for countable quadruple systems

Abstract

From the late 1970s to the early 1980s, Köhler developed a theory for constructing finite quadruple systems with point-transitive Dihedral automorphism groups by introducing a certain algebraic graph, now widely known as the (first) Köhler graph in finite combinatorics. In this paper, we define the countable Köhler graph and discuss countable extensions of a series of Köhler's works, with emphasis on various gaps between the finite and countable cases. We show that there is a simple 2-fold quadruple system over Z with a point-transitive Dihedral automorphism group if the countable Köhler graph has a so-called [1, 2]-factor originally introduced by Kano (1986) in the study of finite graphs. We prove that a simple Dihedral $\ell$-fold quadruple system over Z exists if and only if $\ell = 2$. The paper also covers some related remarks about Hrushovski's constructions of countable projective planes.