4 problems

Let a matroid be a nonempty simplicial complex satisfying the exchange property: whenever $F$ and $G$ are faces with $|F|<|G|$, some $x\in G\setminus F$ has $F\cup\{x\}$ as a face.…

Generator conjecture. The homology of the indecomposable quotient of the simple loopless regular matroid complex is generated by the classes of $U_{1,1}$ and $\mathsf{M}(W_{2k+1})$…

Stanley's realization conjecture. If $\mathbf{h}$ is the $h$-vector of a matroid complex, then there is a pure monomial ideal with Hilbert function $\mathbf{h}$.

Stanley's conjecture. The $h$-vector of a matroid complex is a pure $O$-sequence.