We construct a countably infinite simple rank $3$ matroid $M_*$ which $\wedge$-embeds every finite simple rank $3$ matroid, and such that every isomorphism between finite $\wedge$-subgeometries of $M_*$ extends to an automorphism of $M_*$. We prove that $M_*$ is not $\aleph_0$-categorical, it has the independence property, it admits a stationary independence relation, and that $Aut(M_*)$ embeds the symmetric group $Sym(\omega)$. Finally, we use the free projective extension of $M_*$ to conclude the existence of a countably infinite projective plane embedding all the finite simple rank $3$ matroids and whose automorphism group contains $Sym(\omega)$. 查看全文>>