On axiomatizability of hereditary classes of graphs and matroids

In the paper, hereditary classes of graphs and matroids are studied by means of the model theory. The problems of first-order axiomatizability of some classes of graphs and matroids are considered. The criterion of axiomatizability of monotone hereditary classes of graphs defined by forbidden noninduced subgraphs is proved. Necessary and sufficient conditions of universal and finite axiomatizability of the monotone hereditary classes of graphs are obtained. It is proved that the class of matroids of fixed rank k is finitely axiomatizabile, as well as two hereditary classes of matroids of bounded rank — the class of matroids of rank not exeeding $k$ and the class of partition matroids of rank not exeeding $k$. It is also shown that the hereditary class of matroids of finite rank is nonaxiomatizable.