The “Convexity Conjecture” by Talagrand asks, roughly speaking, whether one can “create convexity” in a bounded number of steps regardless of the dimension of the ambient space. Talagrand also proposed a discrete version of this conjecture, calling it his “lifetime favorite problem” and offering a $1,000 prize for its solution. While the continuous version of the conjecture was recently proven by Hua, Song, and Tudose, the discrete analogue remains wide open. In this talk, we introduce a reformulation of the discrete convexity conjecture using the new notion of “k-thresholds,” an extension of the traditional definition of thresholds. Using this framework, we establish the conjecture for several special cases, focusing primarily on graph properties.

Let $R_k(3)$ denote the smallest integer $N$ such that every $k$-edge-coloring of the complete graph $K_N$ contains a monochromatic triangle. A simple inductive argument gives the classical factorial upper bound $R_k(3)\leq k!=k^{O(k)}$, whereas the best previously known lower bound was only exponential in $k$, namely, $R_k(3)\geq 2^{\Omega(k)}$. It was a longstanding open problem of Erd\H{o}s whether $R_k(3)$ grows exponentially or super-exponentially in $k$. On August 1, 2026, OpenAI, using an internal AI model, discovered a construction establishing the super-exponential lower bound $R_k(3)\geq k^{\Omega(k)}$, thereby resolving Erdős’ longstanding question. In this talk, I will explain the construction and discuss possible directions for further research, some of which may already have been explored by other researchers.

The first major step towards the graph minor structure theorem by Robertson and Seymour was the grid theorem, a result describing that every graph of large treewidth contains a grid as minor. In 2014 Wollan gave a definition for a tree-like decomposition and a width parameter tree-cutwidth with respect to immersions, a different graph containment relation. He provided results linking this parameter to immersions of large walls. This talk presents a version of this parameter for directed graphs, the directed tree-cutwidth. The main result is a grid theorem for directed tree-cutwidth establishing that it is linked to directed immersions of large cylindrical walls. The presented work is joined with Marcin Briański, Karolina Okrasa, and Michał Pilipczuk.

We study restricted-link augmentation to 2-vertex-connectivity. An instance consists of a graph G, possibly disconnected, a set L of admissible links on its vertices, integer link costs in {1, …, W}, and an integer k; the task is to add at most k links of minimum total cost so that the resulting multigraph is 2-vertex-connected. Recent work gives $O^*(k^{O(k)})$-time algorithms for unweighted λ-vertex-connectivity augmentation for every λ ≤ 4 [Carmesin and Ramanujan, SODA 2026], and an $O^*((k + λ)^{O(k)})$-time algorithm for arbitrary λ [Korhonen and Thorup, FOCS 2026]. We give a deterministic algorithm with running time $O^*(36^k W)$. Thus, for λ = 2, the unweighted running time improves from $O^*(k^{O(k)})$ to $O^*(36^k)$, and the algorithm also handles link costs with pseudo-polynomial dependence on W. We reduce the problem to a boundary-pair variant of 2-vertex-connected spanning subgraph, where each vertex is assigned a pair of incident edges with an associated pair cost. We solve this variant using a cancellation identity, inspired by Cut&Count [Cygan et al., TALG 2022], obtained by applying Möbius inversion to decompositions along cut vertices: the identity cancels every connected spanning graph with more than one block and keeps exactly the 2-vertex-connected spanning graphs.

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 $n,\ell$.