Tsukuba Journal of Mathematics

Köhler theory for countable quadruple systems

Hirotaka Kikyo and Masanori Sawa

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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.

Article information

Tsukuba J. Math., Volume 41, Number 2 (2017), 189-213.

Received: 25 May 2016
Revised: 7 September 2017
First available in Project Euclid: 21 March 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05B05: Block designs [See also 51E05, 62K10] 05C63: Infinite graphs 05E18: Group actions on combinatorial structures
Secondary: 05C70: Factorization, matching, partitioning, covering and packing

Hrushovski's construction infinite design $[k − 1,k]$-factor Köhler theory orbit-decomposition projective plane quadruple system


Kikyo, Hirotaka; Sawa, Masanori. Köhler theory for countable quadruple systems. Tsukuba J. Math. 41 (2017), no. 2, 189--213. doi:10.21099/tkbjm/1521597622. https://projecteuclid.org/euclid.tkbjm/1521597622

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