Blocking sets in projective spaces and polar spaces
This article gives a survey on the problem: Consider a finite projective or polar space of dimension \(d\) and two integers \(s\) and \(t\) with \( 0 \leq s,t \leq d\). What is the smallest cardinality of a set \(T\) of \(t\)-subspaces such that every \(s\)-subspace is incident with at least one element of \(T\), and what is the structure of the corresponding set \(T\)? More generally, what are the first few smallest cardinalities of minimal sets \(T\) of \(t\)-subspaces such that every \(s\)-subspace is incident with at least one element of \(T\), and what is the structure of the corresponding sets \(T\)? On this important research problem, many results are known, and still new techniques are being developed to improve on known results and to solve open problems. The author concentrates first of all on results in finite projective spaces. Apart from stating the most important results, he also states, for particular open cases, examples which look very promising to be effectively the smallest examples. As a new result, he presents the solution of the first problem for \(PG(d,q)\) for \(t=1\) and \(2s \leq d \leq 3s-1\). In the final section, he concentrates on results of the first problem, with \(t=0\), in finite classical polar spaces. The author was the first to study this latter problem in great detail, and his research has led to many complete classification results.
scientific article; zbMATH DE number 1972490