Leírás
A well known and widely open question is to describe good criteria for embedding certain set of lines into a finite projective geometry.
Here we propose the following variant of the above question: a set P of points is given as an underlying set, and a set T of triples which are prescribed to be collinear. How does the size of T, or its other combinatorial properties, decide whether T can be realizable by a projective plane of fixed order?
The investigation leads us to several interesting problems concerning linear triple systems, namely two different kinds of 'spreading property' make a key role here. We give lower bounds on the minimal size of linear spreading triple systems applying graph theoretic techniques and upper bounds (constructions) which show connections to subsquare-free Latin squares.
Joint work with Zoltán L. Blázsik