One of the important algorithmic consequences of Robertson and Seymour’s Graph Minor Project is their proof that the k-Vertex-Disjoint Paths problem is fixed-parameter tractable on the class of all undirected graphs, that is, solvable in time $f(k) \cdot n^c$, for some function $f$ and constant $c$. For directed graphs the problem is significantly harder: the k-Disjoint-Paths problem it is NP-complete already for $k=2$. While this indicates that the Directed-k-Disjoint Paths problem is unlikely to be fixed-parameter tractable in general, it is nevertheless interesting to investigate which of the techniques used to solve the problem on undirected graphs fail for digraphs and why and whether some of them can be made to work in a more restricted setting. In this talk I will speak about recent results on disjoint directed paths including positive solutions for special graph classes such as Eulerian digraphs but also recently obtained further hardness results.

For $\boldsymbol{\alpha} = (\alpha_1, \dots, \alpha_k) \in {\mathbb F}_2^k$, an $\boldsymbol{\alpha} $-town is a set family in which every $i$-wise intersection has parity $\alpha_i$. Denote by $f_{\boldsymbol{\alpha} }(n)$ the maximum size of an $\boldsymbol{\alpha} $-town on $[n]$. The classical oddtown and eventown problems study the cases $\boldsymbol{\alpha} = (1, 0)$ and $(0, 0)$, respectively. We determine the sharp asymptotics of $f_{\boldsymbol{\alpha} }(n)$ for all $\boldsymbol{\alpha} $, answering questions of Johnston-O’Neill and Wei-Zhang-Ge.

The cycle double cover conjecture (CDC) claims that every graph without cut-edges has a list of cycles such that every edge appears exactly twice in the list. This conjecture was proposed in 1970s by several mathematicians independently, including Tutte, Seymour, and Szekeres. On July 10, 2026, OpenAI released a proof found by its ChatGPT 5.6 Sol Ultra. I will explain a slightly modified proof, with the aim of making it mostly accessible to undergraduate students.

Let $\alpha(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $\alpha(\mathbb{F}_q^{d},p)$ up to a polylogarithmic factor by proving the balanced supersaturation conjecture of Balogh and Luo. Our result also resolves a conjecture implicitly posed by the first author, Liu, the second author and Zeng. In the course of our proof, we establish a lemma that demonstrates a “structure vs. randomness” phenomenon for point sets in finite-field linear spaces, which may be of independent interest. This is joint work with Jiaxi Nie, Jing Yu, and Wentao Zhang.

A family of sets in $[n]$ is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. The problem was completely solved when $\ell$ is a prime via an elegant linear algebraic method, showing that the family has size at most $n$. However, not much was known for composite numbers. By splitting the family into families correspond to each prime factor of $\ell$, one can show that the number is at most $\omega n$, where $omega$ is the number of prime factors of $\ell$. We used both combinatorial and Fourier analytic arguments to prove that the number of sets in any $\ell$-Oddtown is at most $\omega n-(2\omega+\varepsilon)\log_2 n$ for most $n,\ell$.

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron P whose integer points are precisely those of X. In this paper, we focus on the case where X is the discrete standard simplex $\Delta_d = \{0, e_1, …, e_d\}$. We show that $\operatorname{rc}(\Delta_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(\Delta_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2022) and matches an asymptotic lower bound by Averkov and Schymura (2020). This is joint work with Simon Keil.

The grid theorem of Robertson and Seymour can be equivalently stated using balanced separators, that are separators whose deletion leaves every component with no more than half of the vertices of the graph, as follows. Every graph that excludes some planar graph as a minor has a balanced separator of bounded size. Building on this formulation, Gartland and Lokshtanov conjectured an induced minor version of that theorem inspired by coarse graph theory. They conjectured that every graph that excludes some planar graph as an induced minor has a balanced separator which is dominated by a bounded number of vertices. We confirm this conjecture for excluding any fixed wheel, that is, a cycle together with a universal vertex, as an induced minor. This talk is based on joint work with Maria Chudnovsky, Matjaž Krnc, and Martin Milanič.

This talk deals with induced minor obstructions to treewidth. The natural setup for this problem is to consider the class of graphs excluding some planar graph, and some complete bipartite graph as induced minors, and some complete graph as a subgraph. Unfortunately, such  classes still contain graphs of arbitrarily large treewidth. Moreover, a result of Alecu, Bonnet, Bureo Villafana and Trotignon and its extensions suggest that there is no elegant characterization of families of bounded treewidth in terms of induced obstructions. On the other hand, it is conjectured that graphs in the classes as above have treewidth bounded by a poly-logarithmic function of their number of vertices. If true, this will imply the existence of quasi-polynomial time algorithms for a host of problems on such  classes that are NP-complete in the general setting. While this conjecture remains open, in joint work with Julien Codsi, David Fischer and Daniel Lokshtanov, we were able to prove the existence of a sub-polynomial bound on treewidth in terms of the number of vertices. This in turn leads to sub-exponential algorithmic behavior. In this talk we will discuss some ideas of the proof, and, if time permits, some results in the more general setting when the bound on the clique size is removed.