Online talk: Tony Huynh (CMSA Seminar)

Next week, Tony Huynh is giving a talk in the Combinatorial Mathematics Society of Australasia Seminar which many of our readers may be interested in. Here is the info. The announcement for next week’s Graphs and Matroids talk will follow soon.

Tuesday, October 6, 9pm ET (11am AEST)
Tony Huynh, Monash U
Idealness of k-wise intersecting families
Zoom/password: see CMSA website

 

Abstract:
A clutter is a hypergraph such that no hyperedge is contained in another hyperedge. It is $k$-wise intersecting if every $k$ hyperedges intersect, but there is no vertex contained in all the hyperedges. We conjecture that every $4$-wise intersecting clutter is not ideal. Idealness is an important geometric property, which roughly says that the minimum covering problem for the clutter can be efficiently solved by a linear program. As evidence for our conjecture, we prove it for the class of binary clutters. Our proof combines ideas from the theory of clutters, graphs, and matroids. For example, it uses Jaeger’s $8$-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. We also show that $4$ cannot be replaced by $3$ in our conjecture, where the counterexample of course comes from the Petersen Graph.


This is joint work with Ahmad Abdi, Gérard Cornuéjols, and Dabeen Lee.

Online talk: Dan Cranston

Monday, September 28, 3pm ET (8pm BST, 7am Tue NZST)
Dan Cranston, Virginia Commonwealth U
Vertex Partitions into an Independent Set and a Forest with Each Component Small
Youtube

 

Abstract:
For each integer $k \ge 2$, we determine a sharp bound on $\text{mad}(G)$ such that $V(G)$ can be partitioned into sets $I$ and $F_k$, where $I$ is an independent set and $G[F_k]$ is a forest in which each component has at most $k$ vertices. For each $k$ we construct an infinite family of examples showing our result is best possible. Hendrey, Norin, and Wood asked for the largest function $g(a,b)$ such that if $\text{mad}(G)<g(a,b)$ then $V(G)$ has a partition into sets $A$ and $B$ such that $\text{mad}(G[A])<a$ and $\text{mad}(G[B])<b$. They specifically asked for the value of $g(1,b)$, which corresponds to the case that $A$ is an independent set. Previously, the only values known were $g(1,4/3)$ and $g(1,2)$. We find the value of $g(1,b)$ whenever $4/3<b<2$. This is joint work with Matthew Yancey.

Online talk: Alexey Pokrovskiy (non-standard day & time!)

We’re really excited to have Alexey Pokrovskiy speak on his fantastic recent result on Rota’s Basis Conjecture this upcoming week. This talk has been moved to be a part of the Tutte Colloquium at Waterloo and so will be held at a different day and time than usual. Here are the full details.

Friday, September 25, 1pm ET (6pm BST, 5am Tue NZST)
Alexey Pokrovskiy, Birkbeck, University of London
Rota’s Basis Conjecture holds asymptotically
Youtube

 

Abstract:
Rota’s Basis Conjecture is a well known problem, that states that for any collection of $n$ bases in a rank $n$ matroid, it is possible to decompose all the elements into $n$ disjoint rainbow bases. Here an asymptotic version of this is will be discussed – that it is possible to find $n – o(n)$ disjoint rainbow independent sets of size $n – o(n)$.

Online talk: Oliver Lorscheid

Mon, September 14, 3pm ET (8pm BST, 7am Tue NZST)
Oliver Lorscheid, Instituto Nacional de Matemática Pura e Aplicada
Foundations of Matroids without Large Uniform Minors, Part 2
Youtube

(Part 1 is on Youtube as well)

Abstract:
In this talk, we take a look under the hood of last week’s talk by Matt Baker: we inspect the foundation of a matroid.

The first desired properties follow readily from its definition: the foundation represents the rescaling classes of the matroid and shows a functorial behaviour with respect to minors and dualization. It requires however deep results by Tutte, Dress-Wenzel and Gelfand-Rybnikov-Stone to gain an understanding of the foundation in terms of generators and relations.

For small matroids, this allows us to determine the foundation explicitly. This, in turn, lets us derive the structure theorem for foundations of matroids without large uniform minors, which is the central result behind the applications from last week.

In a concluding part of the talk, we turn to potential future directions and open problems.