6 problems

Let $H$ be an infinite group. A gain-graphic matroid is a frame matroid obtained from a graph with edge gains in $H$, and a class is characterised by a sentence in monadic second-o…

Let $H$ be a finite group. A gain-graphic matroid is a frame matroid obtained from a graph with edge gains in $H$, and a class is well-quasi-ordered when it has neither an infinite…

Let $H$ be a finite group, and let $\mathcal{M}$ be any minor-closed class of $H$-gain-graphic matroids. A gain-graphic matroid is a frame matroid obtained from a graph whose edges…

Let $H$ be a finite group. Efficient pigeonhole conjecture. The class of bicircular matroids is efficiently pigeonhole, and the class of $H$-gain-graphic matroids is efficiently pi…

Let $H$ be an infinite group, and consider the class of rank-$3$ $H$-gain-graphic matroids. Undecidability conjecture. This class has an undecidable ${\mathit{MS}_{0}}$ theory. The…

Let $H$ be an infinite group. Let ${\mathit{CMS}_{0}}$ denote counting monadic second-order logic over matroids without quantification over edge sets. Non-definability conjecture.…