6 problems

Exponential-growth conjecture. The sequence $\lambda_n$ grows at least exponentially in $n$.

Extremal polymatroid $a$-reducedness conjecture. The polymatroid $f$ is actually $a$-reduced; consequently,

Singleton-ratio growth conjecture. The true growth of this supremum is significantly smaller than the doubly exponential upper bound

Let $\mathcal{X}$ be a collection of subsets defining the class of $\mathcal{X}$-matroids, and suppose that this class is nonempty. Let $v_{\mathcal{X}}$ be the function constructe…

Let $G$ be a graph with vertex set $V$, let $c$ be normalized edge weights, and let $\mathcal{S}(G)\subseteq[0,1]^{2^V}$ be the space of resistance capacities. A set function on…

Let $n\geq 2$, let $\boldsymbol{x}\in[0,1]^n$, and let $f:2^{[n]}\rightarrow\mathbb{R}_+$ be a monotone submodular function. Write $f^{+}(\boldsymbol{x})$ for the maximum expected…