Abstract

We give an Ostaszewski-type inductive construction of a locally countable locally compact space which is not α-realcompact but whose onepoint compactification is sequential. This answers a question of Nyikos. The essential ingredient is the use of the Balcar–Vojtas almost-disjoint refinement technique to guide the induction through continuum-many steps.

Introduction

A subset Y of a space X is sequentially closed if no sequence which is a subset of Y converges to a point outside of Y. A space is sequential if each sequentially closed subset is closed. There are not many absolute examples of ‘complicated’ compact sequential spaces in the literature. Furthermore, several important recent results of Balogh, Fremlin and Nyikos, which use Todorčević's ‘forcing positive partition relations’ techniques, show that such spaces cannot be too complicated. For example, they must contain points of first countability and no subspace can be mapped by a closed map onto ω1. The technique, roughly speaking, is to take a countably cpmplete maximal filter of closed sets of a subspace and diagonalize through it with an ω1 sequence that is homogeneous with respect to a certain partition. The homogeneity with respect to the partition guarantees that the sequence ends up being a free sequence in the sense of Arhangel'skii (see [1] or [6]). The upshot is that there cannot be too many countably complete maximal filters on subspaces.

Introduction

Given a fixed graph H on t vertices, a typical graph G on n vertices contains many induced subgraphs isomorphic to H as n becomes large. Indeed, for the usual model of a random graph G* on n vertices (see [4]), in which potential edges are independently included or not each with probability ½, almost all such G* contain induced copies of H as n → ∞. Thus, if a large graph G contains no induced copy of H, it deviates from being ‘typical’ in a rather strong way. In this case, we would expect it to behave quite differently from random graphs in many other ways as well. That this in fact must happen follows from recent work of several authors, e.g., see Chung, Graham & Wilson [5] and Thomason [7], [8]. In this paper we initiate a quantitative study of how various deviations of randomness are related. The particular property we investigate (‘uniform edge density for half sets’ – see Section 3) is just one of many which might have been selected and for which the same kind of analysis could be carried out.

This work also shares a common philosophy with several recent papers of Alon & Bollobás [1] and Erdős & Hajnal [6], which investigate the structure of graphs which have an unusually small number of non-isomorphic induced subgraphs. This is a strong restriction and such graphs must have very large subgraphs which are (nearly) complete or independent.