5 problems

Let $G$ be a graph with edge set $E(G)$, and let an ordering of $E(G)$ determine the monomial lower order ideal $L(G)$ and the ring $R(G)$. An ordering is said to produce an NBC ba…

Let $d,k$ be positive integers. For an ordered matroid $M$, write $\overline{BC_<(M)}$ for its reduced broken circuit complex, and let $h_{d-1}(\overline{BC_<(M)})$ denote its…

Let $M$ be a loopless matroid, and let $(h_0,h_1,\ldots,h_s)$ be the $h$-vector of its broken circuit complex $BC(M)$, where $s$ is the largest index $j$ such that $h_j\ne0$. Flawl…

Let $M$ be a matroid on ground set $E(M)$. A standard ordering of $E(M)$ is an ordering satisfying the standard-ordering conditions defined for the matroid's broken-circuit complex…

Let $M$ be a graphic matroid, meaning the cycle matroid of a graph. An NBC basis conjecture. Every graphic matroid possesses an NBC basis for some standard ordering of its ground s…