I doubt this is true but I was not able to find a clear answer to the question. Surely this is due to my erratic knowledge of matroid theory.
(I know the $U_4^2$-forbidden characterization, but I am unable to make my way from there).
$\begingroup$
$\endgroup$
3
-
3$\begingroup$ Precisely the uniform $U_{2,4}$ is a real representable matroid that is not binary. $\endgroup$Luis Ferroni– Luis Ferroni2023-04-20 17:09:21 +00:00Commented Apr 20, 2023 at 17:09
-
$\begingroup$ @LuisFerroni I see. Are there others? $\endgroup$Felix Goldberg– Felix Goldberg2023-04-20 17:11:24 +00:00Commented Apr 20, 2023 at 17:11
-
2$\begingroup$ Yes, it's easy to construct infinitely many. Take any matrix with real entries having colums of the form $[v_1, v_2, v_3, v_4, v_5,\cdots, v_n]$, where $v_3=v_1-v_2$ and $v_4=v_1+v_2$ and $v_1$ linearly independent of $v_2$. The produced linear matroid is real-representable but it is not binary as it contains a minor isomorphic to $U_{2,4}$ (corresponding to the first four columns). $\endgroup$Luis Ferroni– Luis Ferroni2023-04-20 17:15:31 +00:00Commented Apr 20, 2023 at 17:15
Add a comment
|