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Matroid flat counts can have many peaks
Alexander Divoux, Matt Larson, Chayim Lowen +1
We disprove Rota's conjecture that the counts of flats in a matroid according to rank form a unimodal sequence. Furthermore, we show that this sequence can have arbitrarily many pe…
Matroid flat counts are not unimodal
Alexander Divoux, Chayim Lowen, Shouda Wang
We give counterexamples to Rota's 1970 conjecture that the sequence counting flats of varying rank in a matroid is unimodal. More specifically, inspired by Larson's recent disproof…
Asymptotic structure. V. The coarse Menger conjecture in bounded path-width
Alex Divoux, Tung Nguyen, Alex Scott +1
Menger's theorem tells us that if are sets of vertices in a graph , then (for ) either there are vertex-disjoint paths between and , or there is a set…
Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs
Tantan Dai, Alexander Divoux, Tom Kelly
A hypergraph is \textit{bipartite with bipartition } if every edge has exactly one vertex in , and a matching in such a hypergraph is \textit{-perfect} if it saturate…
Extremal Graphs for the Lights Out Problem
Julien Codsi, Sergio Cristancho, Alexander Divoux +1
Lights Out is a game played on a graph where every vertex has a light bulb that is either on or off, and pressing a vertex toggles the state of every vertex in the closed n…