Description
It is well-known that the Euclidean geometry can be fully constructed by a couple of axioms sorted into 5 groups (some examples of axioms are: any line contains at least two points, two planes never intersect in exactly one point, etc.). However, it is less known that there exist finite models that satisfy all the axioms of the first group, the so-called axioms of incidence. In this talk we discuss some recently discovered classes of such models. Constructions of these models in most cases are based on properties of some discrete structures, such as finite projective and affine spaces, or combinatorial designs. We also analyze some connections between the discovered classes of models (whether they are disjoint, whether one is a subset of another one, etc.). Finally, we determine the exact number of non-isomorphic models with n points up to a certain bound on n (relying, among other things, on, unfortunately, quite a considerable amount of CPU work). This is a joint work with B. Basic, and partly a joint work with (subsets of) M. Maksimovic, N. Miholjcic and M. Sobot.