What is an infinite design?

Cameron, Peter J. and Webb, Bridget S. (2002). What is an infinite design? Journal of Combinatorial Designs, 10(2) pp. 79–91.

DOI: https://doi.org/10.1002/jcd.10005

Abstract

It is usually assumed that an infinite design is a design with infinitely many points. This encompasses a myriad of structures, some nice and others not. In this paper we consider examples of structures that we would not like to call designs, and investigate additional conditions that exclude such anomalous structures. In particular, we expect a design to be regular, the complement of a design to be a design, and a t-design to be an s-design for all 0<s<=t. These are all properties that can be taken for granted with finite designs, and for infintie Steiner systems. We present a new definition of an infinite t-design, and give examples of structures that satisfy this definition. We note that infinite designs considered in the literature to date satisfy this definition. We show that infinite design theory does not always mirror finite design theory, for example there are examples of designs with v>b.

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