COMBINATORIAL PROBLEMS FOR ABELIAN GROUPS ARISING FROM GEOMETRY

Abstract

This paper deals with elementary problems on complexes of abelian groups related to finite geometry, in particular to arcs and blocking sets of finite projective planes. Arcs contained in cubic curves led us to the notion of a 3-independent subset in abelian groups. Various examples of complete arcs containing only three points outside a conic were constructed by KORCHMÁROS [6) using 2 -(m, n) isolated sets. In this paper we survey the known results and constructions concerning 3-independent and 2 (m, n) isolated sets. Moreover we obtain some new bounds for their size and give some new examples showing that the lower and upper bounds are sharp regarding their order of magnitude. Finally, we will show how the methods and constructions of the previous sections can be applied to the problem of blocking sets contained in the union of three lines and answer a question of CAMERON [1].