A classical problem in combinatorial geometry, posed by Erdos in 1946, asks to determine the maximum number of unit segments in a set of $n$ points in the plane. Since then a great variety of extremal problems in finite point sets have been studied. Here, we look at generalizations of this question concerning regular simplices. Among others we answer the following question asked by Erdos: Given $n$ points in $\mathbb{R}^6$, how many triangles can be equilateral triangles? For our proofs we use hypergraph Turan theory. This is joint work with Dumitrescu and Liu.

In this talk, I'll introduce how trees of various flavours arise in the mathematical study of evolution, and briefly describe their applications. Then I will move the focus to just one kind of tree, namely phylogenetic trees. These have uniquely labelled leaves but no other labels, and are the most well-known evolutionary trees (for example, the "tree of life" is a phylogenetic tree). A number of metrics have been introduced on phylogenetic trees, thereby defining a number of (phylogenetic) tree spaces. But what makes a good tree space? I will describe several desirable properties that might make a tree space "good", and compare phylogenetic tree spaces with reference to two desirable properties. I will conclude with remarks on ongoing and emerging challenges.

Sixty years ago, Edmonds published his elegant characterization of a graph $G$'s perfect matching polytope $\mathcal{P}$ (the convex hull of the characteristic vectors of $G$'s perfect matchings). He described $\mathcal{P}$ polyhedrally as the set of nonnegative vectors in $\mathbb{R}^{E(G)}$ satisfying two families of constraints: `saturation' and `blossom'. We now call graphs for which the blossom constraints are essential Edmonds graphs and those for which the blossom constraints are implied by the others Egervary graphs. As it turns out, the second graph class interacts with more familiar ones in satisfying ways. For example, bipartite graphs are Egervary as are Konig-Egervary graphs. This talk introduces these graph classes and shares a few results on Egervary graphs that appeared recently in our Journal of Combinatorics article. (Joint work with Jack Edmonds and Craig Larson)

Lightning Talks

Richard Hoshino, Northeastern University

Creating Mathematically-Optimal Timetables for Schools

Self-dual binary linear codes have been extensively studied and classified for length $n\leq 40$. However, little attention has been paid to linear codes that coincide with their orthogonal complement when the underlying inner product is not the dot product. We introduce an alternating form defined on $\mathbb{F}_2^n$ and study codes that are maximal totally isotropic with repsect to this form. We classify such codes for $n\leq 24$ and present a MacWilliams-type identity which relates the weight enumerator of a linear code and that of its orthogonal complement with respect to our alternating inner product. As an application, we derive constraints on the weight enumerators of maximal totally isotropic codes.

The FKG inequality is a powerful tool for proving inequalities in distributive lattices. We show how a special case, which we call the Order Ideal Lemma, can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner. We use the Order Ideal Lemma to prove log-concavity and log-convexity of various sequences involving lattice paths, intervals in Young’s lattice, order polynomials, Lucas sequences, descent and peak polynomials of permutations, pattern avoidance, set partitions, and noncrossing partitions. This is a joint work with Bruce Sagan.

Ramsey’s remarkable theorems demonstrate the presence of order inside large chaotic systems. They have proven useful in a wide range of mathematical topics, including discrete mathematics, number theory, and computer science. However, it is often forgotten that F.P. Ramsey was motivated by the topic of mathematical logic, specifically Hilbert’s Entscheidungsproblem, and that the proofs of the infinite and finite Ramsey theorems took up but 6 pages of Ramsey’s 23 page paper On a Problem in Formal Logic. In my talk, I will invite others to reflect with me on the underappreciated origin of this beautiful theory.

Cops and Robbers is a pursuit--evasion game played on graphs. In the damage version of the game, the robber damages each distinct vertex he visits, unless immediately captured by the cop. We define the ${\textit damage\ number}$ of a graph to be the minimum number of vertices that can be damaged by the robber, and we say that a graph is $\textit{cop-win}$, if the cop always has a strategy to capture the robber. It has been shown that the damage number of a cop--win graph of order $n\ge 7$ is at most $n-5$, and tight upper bounds have been established for the damage numbers of graphs of order $n\le9$. In this talk, we extend these results by identifying structural properties that, when present, guarantee a damage number of at most $n-5$ for all graphs $n\ge10$.