3 problems

White's $TE$ conjecture. All matroids satisfy each of the three properties:

Let $M$ be a matroid, with toric ideal $I_M$. For bases $B_1,B_2,D_1,D_2$, call $y_{B_1}y_{B_2}-y_{D_1}y_{D_2}$ a double-swap binomial when the pair $D_1,D_2$ is obtained from…

Let $G$ be a unicyclic graph on $n$ vertices, and let $\mathfrak{c} \in \mathbb{Z}_{>0}^n$. Write $\operatorname{Ker}(\pi_G^{\mathfrak{c}})$ for the associated toric ideal, and cal…