Packing Plane Spanning Trees and Paths in Complete Geometric Graphs

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in $\Omega(\sqrt{n})$. Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).Comment: This work was presented at the 26th Canadian Conference on Computational Geometry (CCCG 2014), Halifax, Nova Scotia, Canada, 2014. The journal version appeared in Information Processing Letters, 124 (2017), 35--4

Topological dynamics beyond Polish groups

When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat generalization of this dividing line to all topological groups. We prove a number of characterizations of this class, having very different flavors, and use these to prove that the class of CAP groups enjoys a number of nice closure properties. As a concrete application, we compute the universal minimal flow of the homeomorphism groups of several scattered topological spaces, building on recent work of Gheysens

Set Theory

This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject

Avoiding, pretending, and querying : three combinatorial problems.

A k-term quasi-progression of diameter d is a sequence {Xl,... ,xk} for which there exists a positive integer l such that l \u3c Xi-Xi-1 \u3c l+d, for all i = 2, ... ,k. Quasi-progressions may be thought of as arithmetic progressions with a certain amount of \u27wiggle-room\u27 allowed. Let Q(d, k) be the least positive integer such that every 2-coloring of {1, ... , Q(d, k)} contains a monochromatic k-term quasi-progression of diameter d. We prove a conjecture of Landman for certain values of k and d, provide counterexamples for some other cases, and determine that the conjecture always has the correct order of growth. Let A be the adjacency matrix of a non empty graph. Is there always a nonzero {0, 1}-vector in the row space of A that is not a row of A? Akbari, Cameron, and Khosrovshahi have shown that an affirmative answer to this question would imply bounds on many graph parameters as a function of the rank of the adjacency matrix. We demonstrate the existence of such vectors for certain families of graphs, examine techniques to find and verify the existence of such vectors, and show that if you generalize the problem to allow asymmetry in the matrices then some {0, 1 }-matrices do not have such vectors. In 1981, Andrew Yao asked Should tables be sorted? . When the table has n cells that are filled with entries taken from a key space of m possibilities, he showed that it is possible to decide whether any member of the key space is present in the table by inspecting (querying) only one cell of the table if and only if m \u3c 2n - 2. We make steps toward extending his result to the case where you are permitted two queries by considering several variations of the problem