Online Talk: Nathan Bowler

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Please advertise this talk at your home institution. Anyone is welcome to attend! 

Time: Wednesday, Sep 20 at 3pm ET

Speaker: Nathan Bowler, Universität Hamburg
Title: The $K_4$ game

Abstract: Two players alternately claim edges of a complete graph on infinitely many vertices. The first to claim all edges of a $K_4$ wins. Can the first player force a win? I will explain the history of this question, why it is harder than it seems, and how to win this game.

Online Talk: Kevin Grace

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Time: Thursday, July 20, 3pm ET

Speaker: Kevin Grace, Vanderbilt University
Title: Some generalizations of the class of spikes

Abstract: Spikes (also called tipless spikes in the matroid theory literature) form a well-known class of matroids that are important in the study of matroid connectivity. These matroids have the property that every pair of elements is contained in both a 4-element circuit and a 4-element cocircuit. We will present a family of generalizations of spikes, which we call (s, t)-spikes, with the property that every s-element subset of the ground set is contained in a 2s-element circuit and every t-element subset of the ground set is contained in a 2t-element cocircuit. We call this property the (s, 2s, t, 2t)-property. Our main result is that all sufficiently large matroids with the (s,2s, t, 2t)-property are (s, t)-spikes. This is joint work with Nick Brettell.

Online Talk: George Drummond

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Time: Thursday, July 6, 3pm ET

Speaker: George Drummond, University of Canterbury
Title: Elastic elements and where to find them

Abstract: In this talk we will discuss the presence of “elastic” elements in 3-connected matroids. An element e of a 3-connected matroid M is elastic if both the simplification of M/e and the cosimplification of M\e are 3-connected. Those familiar with Bixby’s Lemma will recall that at least one of these two conditions holds for every element. We will consider the obstructions to the presence of elastic elements, namely 4-element fans and another family of 3-separating structures we dub “theta-separators”, before arriving at our wheels-and-whirls-type theorem for elastic elements. In particular, it is shown that if neither of the aforementioned obstructions is present, then we are guaranteed at least four elastic elements. This wheels-and-whirls-type result then extends naturally to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified 3-connected minor. We will also consider some applications of these results to the related study of removing elements relative to a fixed basis.

Online Talk: Tung Nguyen

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Time: Thursday, June 22, 3pm ET

Speaker: Tung Nguyen, Princeton University
Title: More graphs with the Erdős–Hajnal property

Abstract: Erdős and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or stable set of size at least $n^c$. Alon, Pach, and Solymosi reduced this conjecture to the case when $H$ is prime, or equivalently when $H$ is not a blow-up of smaller graphs. Until now, it was not known for any prime $H$ on at least six vertices. This talk describes a construction of infinitely many prime graphs $H$ each satisfying the Erdős–Hajnal conjecture. The proof method actually gives the stronger result that every such $H$ is viral, which will be explained in the talk. Joint work with Alex Scott and Paul Seymour.