Abstract: The minimum bisection width of random cubic (3-regular) graphs is of interest because it is one of the simplest questions imaginable in extremal combinatorics. An additional reason is that the minimum bisection of (general) cubic graphs plays a role in the construction of efficient exponential-time algorithms, and it seems likely that random cubic graphs are extremal.
It is known that a random cubic graph has a minimum bisection of size at most 1/6 times its order (indeed this is known for all cubic graphs), and we reduce this upper bound to below 1/7 (to 0.13993) by analyzing an algorithm with a couple of surprising features. We increase the corresponding lower bound on minimum bisection using the Hamilton cycle model of a random cubic graph. (The Hamilton cycle approach had also decreased the upper bound on maximum cut, but this has since been improved upon by making rigorous a 2010 conjecture from statistical physics.)

A Universal Construction for Unique Sink Orientations

Abstract: The minimum bisection width of random cubic (3-regular) graphs is of interest because it is one of the simplest questions imaginable in extremal combinatorics. An additional reason is that the minimum bisection of (general) cubic graphs plays a role in the construction of efficient exponential-time algorithms, and it seems likely that random cubic graphs are extremal.
It is known that a random cubic graph has a minimum bisection of size at most 1/6 times its order (indeed this is known for all cubic graphs), and we reduce this upper bound to below 1/7 (to 0.13993) by analyzing an algorithm with a couple of surprising features. We increase the corresponding lower bound on minimum bisection using the Hamilton cycle model of a random cubic graph. (The Hamilton cycle approach had also decreased the upper bound on maximum cut, but this has since been improved upon by making rigorous a 2010 conjecture from statistical physics.)

Abstract: Suppose we weight the edges of the complete graph K_n with independent exponential Exp(1) random weights, pick two distinct vertices s and t,and then successively construct and then remove the edges of minimal weight s-t paths. We describe asymptotically the distributions of the weights of the first k paths obtained in this process. In particular we show that the mean weight of the k-th path is

(log n + γ + 2ζ(3) + 2ζ(5) + ... + 2ζ(2k − 1) + o(1))/n

as n →  when k is a constant, and where γ is the Euler--Mascheroni constant and ζ is the Riemann zeta function.

This is joint work with P. Balister.

Abstract: In recent years, the theory of pseudorandom graphs have received considerable attention in combinatorial research. One question raised was, how pseudorandom K_s-free graphs can be. A note by Mubayi and Verstraëte published in 2019 describes a link between pseudorandom graphs and Ramsey numbers: They show that if pseudorandom K_s-free graphs exist, then the Ramsey number r(s,t) can be bounded from below. In this talk, we will look at this note. We will try to understand the link in detail. Could pseudorandom graphs really be the "the central tool required to determine classical graph Ramsey numbers" as the authors suggest?

Abstract: The Erdős-Hajnal Conjecture is one of the most well-known problems in extremal combinatorics. It states that in contrast to the general n-vertex graph, if one forbids any fixed graph H as an induced subgraph, instead of a tight guarantee of a logarithmic-size homogeneous set, one can guarantee a homogeneous set of polynomial size. While the conjecture is far from being fully resolved, it is known to be true for some forbidden subgraphs of small order. The smallest forbidden subgraph for which the conjecture remains open is the path on five vertices. In this talk I intend to review a recent paper by Blanco and Bucić guaranteeing a homogeneous set of size 2^{Ω(log(n)^2/3)} in P₅-free graphs. This result is an improvement to the 2^{Ω(log(n)^1/2)} size of a homogeneous set guaranteed by a result of Erdős and Hajnal for any forbidden subgraph.

Abstract: Consider the following model of random graphs: a total of n vertices are assigned to uniformly random positions on the unit square, independently of each other, and any two vertices are then joined by an edge if the distance between their positions is less than a given parameter r. This is called the random geometric graph G(n,r) and, similarly to the binomial random graph G(n,p), increasing properties exhibit thresholds (with respect to the parameter r) which we wish to understand. The behaviour of random geometric graphs, however, is very different from the behaviour of G(n,p).

In this talk, I will highlight some of these differences and eventually focus on the case of balanced s-ary trees, for which we have established the threshold. This is based on joint work with Lyuben Lichev, Dieter Mitsche and Alexandra Wesolek.

Abstract: An essential cover of the vertices of the n-cube {-1,1}ⁿ by hyperplanes is a minimal covering where no hyperplane is redundant and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a construction of an essential cover with n/2+1 hyperplanes and showed that √n hyperplanes are required. Recently, Yehuda and Yehudayoff improved the lower bound by showing that any essential cover of the n-cube contains at least n^0.52 hyperplanes. In this talk, we show that n^5/9 hyperplanes are needed. This is based on a joint work with Araujo and Balogh.

Abstract: Flag Algebras is a highly successive formalism to study problems in extremal combinatorics. It has achieved the best known bounds for Turán’s (3,4)-problem as well as tight bounds for the three color Ramsey multiplicity of the triangle, Erdös’s Pentagon conjecture and a conjecture of Erdös and Sós on the asymptotic frequency of Rainbow triangles. What is special about Flag Algebras is that a significant amount of work in finding a Flag Algebra proof certificate can be automated by relaxing the given problem to a semidefinite program (SDP).

In this talk, we will formally define Flag Algebras with many examples from the theory of c-edge-colored complete graphs. We will then see how to set up a Flag Algebra SDP and will discuss methods, old and new, to reduce the size of the SDP as well as splitting it up into smaller blocks. Finally, we will present our newest calculation for the four color Ramsey multiplicity of the triangle.

This is joint work with Prof. Pokutta (TU Berlin and ZIB) and Dr. Christoph Spiegel (ZIB).

Abstract: Given a collection of m hypergraphs on the same vertex set, an m-edge graph F is a transversal if there is each edge comes from a different graph of the collection. How large does the minimum degree of each graph in the collection need to be so that it necessarily contains a copy of F that is a transversal? Each graph in the collection could be the same hypergraph, hence the minimum degree of each of them needs to be large enough to ensure that it individually contains F. In this talk, we discuss a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles.

This is joint work with Pranshu Gupta, Fabian Hamann, Alp Müyesser, and Amedeo Sgueglia.

Abstract: The lattice width of a convex body measures how "thin" the body is in lattice directions. The so-called flatness theorem states that in each fixed dimension there is an upper bound for the lattice width of a special class of convex bodies, those which are hollow. In this talk we introduce these definitions and certain generalizations thereof and look at the few known extremal hollow convex bodies achieving maximum lattice width. This leads us to a conjecture about extremal convex bodies, namely that they are lattice-reduced. This class of convex bodies has not been previously studied, and we present (many) questions and (some) answers about such lattice-reduced convex bodies. 

Abstract: Given graphs H and G the H-bootstrap process on G is the process that starts with G and in which at each step every edge that is the only missing edge in a copy of H is added. Any such process stabilises after a finite number of steps. The maximum running time M_H(n) is the largest number of steps of an H-bootstrap process when H is fixed and G ranges over all graphs on n vertices. In this talk we investigate M_H(n) when H is distributed as the random graph G(k,p). We will show that for p = ω(log(k)/k) the maximum running time is bounded from below by c_kn² for some c_k>0.

This is joint work with Patrick Morris and Tibor Szabó.

Abstract: A Unique Sink Orientation (USO) is an orientation of the hypercube graph, such that the induced subgraph of every non-empty subcube contains exactly one sink. USOs can be used to capture the combinatorial structure of many essential algebraic and geometric problems. Unfortunately, there is no known algorithm that verifies if a given orientation is actually USO and which runs in polynomial time in the dimension of the cube. Out of the many possible orientations for a given cube only few of them are USO. But still, for a cube of a fixed dimension, the number of USOs is exponential in the dimension. To generate bounds on the total number of USOs, construction techniques are needed. They are also useful to find counterexamples to suspected properties of USOs. Furthermore, families of ''bad'' USOs provide examples to show lower bounds for the worst-case runtime of algorithms. While there are some construction methods for USOs, until now they have not been able to generate all possible USOs.

In this talk we present a new construction framework which can be applied to all USOs and with which we can generate every n-dimensional USO using only USOs of dimension n-1 or lower. Our universal construction was inspired by techniques from cube tilings of space.

This is joint work with Joseph Doolittle (TU Graz) and Simon Weber (ETH Zürich)