Fedor Noskov, Polynomial dependencies in hypergraph Turan-type problems
2025-11-18 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
Consider a general Turan-type problem on hypergraphs. Let $mathcal{F}$ be a family of $k$-subsets of $[n]$ that does not contain sets $F_1, ldots, F_s$ satisfying some property $P$. We show that if $P$ is low-dimensional in some sense (e.g., is defined by intersections of bounded size) then, under polynomial dependencies between $n, k$ and the parameters of $P$, one can reduce the problem of maximizing the size of the family $|mathcal{F}|$ to a finite extremal set theory problem independent of $n$ and $k$. We show that our technique implies new bounds in a number of Turan-type problems including the Erdős-Sós forbidden intersection problem, the Duke-Erdős forbidden sunflower problem, forbidden $(t, d)$-simplex problem and the forbidden hypergraph problem. Furthermore, we also briefly discuss the connection between the aforementioned reduction and the measure boosting argument based on the action of a certain semigroup on the Boolean cube. This connection turns out to be fruitful when extending extremal set theory problems to domains different from $binom{[n]}{k}$.
Joint work with Liza Iarovikova, Andrey Kupavskii, Georgy Sokolov and Nikolai Terekhov

Péter Pál Pach, Product representation of perfect cubes
2025-11-25 16:30 - 17:30
Let $F_{k,d}(n)$ be the maximal size of a set ${A}subseteq [n]$ such that the equation has no solution with $a_1,a_2,ldots,a_kin A$ and integer $x$. Erdős, Sárközy and T. Sós studied $F_{k,2}$, and gave bounds when $k=2,3,4,6$ and also in the general case. We study the problem for $d=3$, and provide bounds for $k=2,3,4,6$ and $9$, furthermore, in the general case, as well. In particular, we refute an 18-year-old conjecture of Verstraëte.
We also introduce another function $f_{k,d}$ closely related to $F_{k,d}$: While the original problem requires $a_1, ldots , a_k$ to all be distinct, we can relax this and only require that the multiset of the $a_i$’s cannot be partitioned into $d$-tuples where each $d$-tuple consists of $d$ copies of the same number.
Joint work with Fleiner, Juhász, Kövér and Sándor.

Matthew Kwan, Exponential anticoncentration of the permanent
2025-12-08 16:30 - 17:30
Let A be a random n×n matrix with independent entries, and suppose that the entries are “uniformly anticoncentrated” (for example, A could be a uniformly random n×n matrix with ±1 entries). We prove that the permanent of A is exponentially anticoncentrated, significantly improving previous bounds of Tao and Vu. Our proof also works for the determinant, giving an alternative proof of a classical theorem of Kahn, Komlós and Szemerédi. Joint work with Zach Hunter and Lisa Sauermann.

Tuukka Korhonen, Dynamic Treewidth in Logarithmic Time
2025-12-09 16:30 - 17:30
We present a dynamic data structure that maintains a tree decomposition of width at most 9k+8 of a dynamic graph with treewidth at most k, which is updated by edge insertions and deletions. The amortized update time of our data structure is $2^{O(k)} log n$, where n is the number of vertices. The data structure also supports maintaining any “dynamic programming scheme” on the tree decomposition, providing, for example, a dynamic version of Courcelle’s theorem with $O_k(log n)$ amortized update time; the $O_k(⋅)$ notation hides factors that depend on k. This improves upon a result of Korhonen, Majewski, Nadara, Pilipczuk, and Sokołowski [FOCS 2023], who gave a similar data structure but with amortized update time $2^{k^{O(1)}} n^{o(1)}$. Furthermore, our data structure is arguably simpler. Our main novel idea is to maintain a tree decomposition that is “downwards well-linked”, which allows us to implement local rotations and analysis similar to those for splay trees.
This talk is based on arXiv:2504.02790.

Chi Hoi Yip, Cliques in Paley graphs and cyclotomic graphs
2025-12-16 16:30 - 17:30
Given a prime power $q equiv 1 pmod 4$, the Paley graph of order $q$ is the graph defined over $mathbb{F}_q$ (the finite field with $q$ elements), such that two vertices are adjacent if and only if their difference is a square in $mathbb{F}_q$. In this talk, I will present some recent progress on the clique number of Paley graphs of non-square order, the characterization of maximum cliques in Paley graphs of square order, as well as their extensions to cyclotomic graphs. In particular, I will highlight a new proof of the Van Lint–MacWilliams’ conjecture using ideas from arithmetic combinatorics.

Donggyu Kim (김동규), Grassmann-Plücker functions for orthogonal matroids
2025-12-26 16:30 - 17:30
We present a new cryptomorphic definition of orthogonal matroids with coefficients using Grassmann-Plücker functions. The equivalence is motivated by Cayley’s identities expressing principal and almost-principal minors of a skew-symmetric matrix in terms of its pfaffians. As a corollary of the new cryptomorphism, we deduce that each component of the orthogonal Grassmannian is parameterized by certain part of the Plücker coordinates.
This is joint work with Changxin Ding.

Yunbum Kook (국윤범), Sampling and volume computation
2025-12-30 16:30 - 17:30
Since the development of the first randomized polynomial-time algorithm for volume computation by Dyer, Frieze, and Kannan in 1989, convex-body sampling has been a central problem at the intersection of algorithms, geometry, and probability. A major milestone came in 1997, when Kannan, Lovász, and Simonovits analyzed the Ball Walk and formulated the influential KLS conjecture. This was extended to log-concave distributions by Lovász and Vempala in 2006, and further accelerated by Cousins and Vempala in 2015 through warm-start generation techniques.
In this talk, I will present a gentle introduction to these milestones and how they have been streamlined and improved in the past few years. The talk is based on joint work with Santosh Vempala.

Daniel Mock, A Simple Algorithm for the Dominating Set Problem and More
2026-01-06 16:30 - 17:30
In [1], Fabianski et. al. developed a simple, yet surprisingly powerful algorithmic framework to develop efficient parameterized graph algorithms. Notably they derive a simple parameterized algorithm for the dominating set problem on a variety of graph classes, including powers of nowhere dense classes and biclique-free classes. These results encompass a wide range of previously known results and often improve the best known running times. Similar results follow for the distance-r variation of dominating set and for independent set. The running time of the algorithm is closely tied to model-theoretic properties, i.e. stability and the Helly property.
We build upon these results and develop a similar algorithm which only relies on the strong Helly property and does not need stability. For the dominating set problem, we get a parameterized algorithm that works (additionally to biclique-free classes and powers of nowhere dense classes) weakly gamma-closed classes while being simpler and faster than previously known results.
In this talk, we introduce the basic framework, results by Fabianski et. al and connections to other areas. We discuss our new insights and possible research directions.
[1] Grzegorz Fabianski, Michal Pilipczuk, Sebastian Siebertz, Szymon Torunczyk: Progressive Algorithms for Domination and Independence. STACS 2019

Ferdinand Ihringer, Boolean Functions Analysis in the Grassmann Graph
2026-01-13 16:30 - 17:30
Boolean function analysis for the hypercube ${ 0, 1 }^n$ is a well-developed field and has many famous results such as the FKN Theorem or Nisan-Szegedy Theorem. One easy example is the classification of Boolean degree $1$ functions: If $f$ is a real, $n$-variate affine function which is Boolean on the $n$-dimensional hypercube (that is, $f(x) in { 0, 1 }$ for $x in { 0, 1 }^n$), then $f(x) = 0$, $f(x) = 1$, $f(x) = x_i$ or $f(x) = 1 – x_i$. The same classification (essentially) holds if we restrict ${ 0, 1}^n$ to elements with Hamming weight $k$ if $n-k, k geq 2$. If we replace $k$-sets of ${ 1, ldots, n }$ by $k$-spaces in $V(n, q)$, the $n$-dimensional vector space over the field with $q$ elements, then suddenly even the simple question of classifying Boolean degree $1$ functions, here traditionally known as Cameron-Liebler classes, becomes seemingly hard to solve.
We will discuss some results on low-degree Boolean functions in the vector space setting. Most notably, we will discuss how vector space Ramsey numbers, so extremal combinatorics, can be utilized in this finite geometrical setting.

Tomáš Masařík, Separator Theorem for Minor-free Graphs in Linear Time
2026-01-20 16:30 - 17:30
The planar separator theorem by Lipton and Tarjan [FOCS ’77, SIAM Journal on Applied Mathematics ’79] states that any planar graph with $n$ vertices has a balanced separator of size $O(sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan’s theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC ’90, Journal of the AMS ’90] showed that any minor-free graph admits a balanced separator of size $O(sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(sqrt{n})$ or have superlinear running time, or both.
In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.
This is a joint work with Édouard Bonnet, Tuukka Korhonen, Hung Le, and Jason Li.

Daniel Dadush, A Strongly Polynomial Algorithm for Linear Programs with at Most Two Non-Zero Entries per Row or Column
2026-01-27 16:30 - 17:30
We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Végh (MOR ’17) and Megiddo (SICOMP ’83) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale’s 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS ’22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL ’04), the same bound applies to any linear program with at most two non-zeros per column or per row
Joint work with Zhuan Khye Koh (Boston U), Bento Natura (Columbia), Neil Olver (LSE), and László A. Végh (Bonn).

Xiaofan Yuan (袁晓璠), Rainbow structures in edge colored graphs
2026-02-03 16:30 - 17:30
Let $G = (V, E)$ be a graph on $n$ vertices, and let $c : E to P$, where $P$ is a set of colors. Let $delta^c(G) = min_{v in V} { d^{c}(v) }$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $delta^c(G)geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. We show that for sufficiently large graphs $G$, the same bound for $delta^c(G)$ implies that any two vertices are connected by a rainbow path. We also show sufficient conditions on $delta^c(G)$ for the existence of a rainbow cycle of length $2k$ in sufficiently large bipartite graphs $G$, which are tight in many cases. This is joint work with Andrzej Czygrinow.

Seonghun Park (박성훈), Formalizing Flag Algebras in the Lean Theorem Prover
2026-02-10 16:30 - 17:30
Flag algebras are a mathematical framework introduced by Alexander Razborov in 2007, which has been used to resolve a wide range of open problems in extremal graph theory in the past twenty years. This framework provides an algebraic setup where one can express relationships between induced subgraph densities symbolically. It also comes with mathematical techniques for systematically deriving such relationships that always hold. Some of these techniques have even been implemented in automatic tools, such as Flagmatic. In this work, we formalise flag algebras in Lean, an interactive theorem prover based on dependent type theory. Lean is computer software that lets us write mathematical definitions and proofs in a computer and check the correctness of written proofs using a computer. By formalizing flag algebras in Lean, we can ensure that the mathematical foundation of flag algebras does not have any mistakes, and provide a mathematical guarantee that theorems proved by flag algebras are indeed correct. In this talk, I will explain flag algebras and our Lean formalization using Mantel’s theorem as a guiding example. In the process, I will describe the challenges and lessons learned during the formalization.
This is a joint work with Jihoon Hyun, Gyeongwon Jeong, Sang-il Oum, and Hongseok Yang.

Marek Sokołowski, Strongly Polynomial Parallel Work-Depth Tradeoffs for Directed SSSP
2026-02-24 16:30 - 17:30
In this talk, we show new strongly polynomial work-depth tradeoffs for computing single-source shortest paths (SSSP) in non-negatively weighted directed graphs in parallel. Most importantly, we prove that directed SSSP can be solved within $widetilde{O}(m+n^{2-varepsilon})$ work and $widetilde{O}(n^{1-varepsilon})$ depth for some positive $varepsilon>0$. For dense graphs with non-negative real weights, this yields the first nearly work-efficient strongly polynomial algorithm with sublinear depth. Moreover, we develop efficient parallel algorithms in the Word RAM model for several variants of SSSP in graphs with exponentially large edge weights.
This is a joint work with Adam Karczmarz and Wojciech Nadara.

Chính T. Hoàng, Problems on graph coloring
2026-03-03 16:30 - 17:30
A k-coloring of a graph is an assignment of k colors to its vertices such that no two adjacent adjacent vertices receive the same color. The Coloring Problem is the problem of determining the smallest k such that the graph admits a k-coloring. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. The complexity of the Coloring Problem for L-free graphs is known (NP-complete or polynomial-time solvable) whenever L contains a single graph. There has been keen interest in coloring graphs whose forbidden list L contains basic graphs such as induced paths, induced cycles and their complements. In this talk, I will provide a survey of recent progress on this topic.

Dario Cavallaro, Well-quasi-ordering Eulerian directed graphs by (strong) immersion
2026-03-10 16:30 - 17:30
Directed graphs prove to be very hard to tame in contrast to undirected graphs. In particular, they are not well-quasi-ordered by any known relevant inclusion relation, and are lacking fruitful structure theorems. This motivates the search for structurally rich subclasses of directed graphs that are well behaved. Eulerian directed graphs are a particularly prominent example, sharing many similarities with undirected graphs. In fact, it is conjectured that Eulerian directed graphs are well-quasi-ordered by weak immersion, and even well-quasi-ordered by strong immersion when restricting to classes of bounded degree. We believe that we have a proof of both conjectures, and I will report on the current status, progress, and steps towards said proof and its implications. This is joint work with Ken-ichi Kawarabayashi and Stephan Kreutzer.

József Balogh, Clique covers and decompositions of cliques of graphs
2026-03-12 16:30 - 17:30
Two related papers will be discussed:
1. In 1966, Erdős, Goodman, and Pósa showed that if $G$ is an $n$-vertex graph, then at most $lfloor n^2/4 rfloor$ cliques of $G$ are needed to cover the edges of $G$, and the bound is best possible as witnessed by the balanced complete bipartite graph. This was generalized independently by Győri–Kostochka, Kahn, and Chung, who showed that every $n$-vertex graph admits an edge-decomposition into cliques of total `cost’ at most $2 lfloor n^2/4 rfloor$, where an $i$-vertex clique has cost $i$. Erdős suggested the following strengthening: every $n$-vertex graph admits an edge-decomposition into cliques of total cost at most $lfloor n^2/4 rfloor$, where now an $i$-vertex clique has cost $i-1$. We prove fractional relaxations and asymptotically optimal versions of both this conjecture and a conjecture of Dau, Milenkovic, and Puleo on covering the $t$-vertex cliques of a graph instead of the edges. Our proofs introduce a general framework for these problems using Zykov symmetrization, the Frankl–Rödl nibble method, and the Szemerédi Regularity Lemma. It is joint work with Jialin He, Robert Krueger, The Nguyen, and Michael Wigal.
2. Let $r ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Győri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Turán graph on $n$ vertices and $r – 1$ parts, then $G$ has at least $(2 – o(1))k/r$ edge-disjoint $r$-cliques. We prove this conjecture. It is joint work with Michael Wigal.

Hidde Koerts, Characterizing large clique number in tournaments
2026-03-24 16:30 - 17:30
A backedge graph of a tournament $T$ with respect to a total ordering $prec$ of the vertices of $T$ is a graph on $V(T)$ where $uv$ is an edge if and only if $uv in A(T)$ and $v prec u$. In 2023, Aboulker, Aubian, Charbit and Lopes introduced the clique number of tournaments based on backedge graphs as a natural counterpart to the dichromatic number of tournaments. Specifically, the clique number of a tournament is the minimum clique number of a backedge graph when considering all possible orderings.
Given this definition, it is not immediately clear what the canonical clique object should be. In this talk, we provide an answer to this question. We show that if a tournament has large clique number, it contains a reasonably large subtournament from one of two simple and previously studied families of tournaments of unbounded clique number.
This talk is based on joint work with Logan Crew, Xinyue Fan, Benjamin Moore, and Sophie Spirkl.

Tung H. Nguyen, Polynomial χ-boundedness for excluding the five-vertex path
2026-03-31 16:30 - 17:30
We overview the recent resolution of a 1985 open problem of Gyárfás, that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path. The proof introduces a chromatic density framework involving chromatic quasirandomness and chromatic density increment, which allows us to deduce the desired statement from the Erdős–Hajnal result for the five-vertex path.

Mamadou Moustapha Kanté, Strongly flip-flat classes of graphs
2026-04-07 16:30 - 17:30
Strong flip-flatness appears to be the analogue of uniform almost-wideness in the setting of dense classes of graphs. Almost-wideness is a notion that was central in different characterisations of nowhere dense classes of graphs, and in particular the game-theoretic one. In this talk I will present the flip-flatness notions and conjectures about the characterization of strongly flip-flat graph classes. Then, I will present a proof that strongly flip-flat classes of graphs that are weakly sparse are indeed uniformly almost-wide, making a step towards their characterisation. A consequence is a characterization of strongly flip-flat graph classes with low rank-depth colourings.
This is a joint work with F. Ghasemi, J. Grange and F. Madelaine.

Xin Wei, Separating hash families with large universe
2026-04-28 16:30 - 17:30
Separating hash families are useful combinatorial structures that generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of the universe for a $t$-perfect hash family of length $N$ over an alphabet of size $q$. We show that $q^{2 – o(1)} < p_t(t, q) = o(q^2)$ for all $t ge 3$, thereby resolving an open problem raised by Blackburn et al. (2008) for certain parameter ranges. Previously, this result was known only for $t = 3$ and $t = 4$. Our approach establishes the existence of a large set of integers that avoids nontrivial solutions to a system of correlated linear equations. This is joint work with Xiande Zhang and Gennian Ge.

Maximilian Gorsky, The Disjoint Paths Problem lies in the Oort cloud of algorithms
2026-05-06 16:30 - 17:30
In this talk we discuss recent work to that establishes that the bounds of the Vital Linkage Function is single-exponential. This has immediate impacts on the complexity of the k-Disjoint Paths Problem, Minor Checking, and more generally, the Folio-Problem. We in fact prove something even stronger: It turns out that it is not in fact the number of terminals (or more generally vertices) that matters in these problems, but rather their structure within the graph. Concretely, we show that the Vital Linkage Function is single-exponential only in the bidimensionality of the terminals, whilst the number of terminals contributes only polynomially. A direct consequence of this is an algorithm for the k-Disjoint Paths Problem running in $f(k)n^2$-time, where f(k) is singly exponential in k and doubly exponential in the bidimensionality of k. This derives directly from an algorithm for the Folio-problem we give that has an analogous runtime. Notably these are the first algorithms for these problems in which the function f is explicit. In particular, we give the first explicit bounds for the Vital Linkage Function.
Joint work with Dario Cavallaro, Stephan Kreutzer, Dimitrios Thilikos, and Sebastian Wiederrecht.

Benjamin Duhamel, Excluding a forest induced minor
2026-05-12 16:30 - 17:30
We give an induced counterpart of the Forest Minor theorem: for any t ≥ 2, the $K_{t,t}$-subgraph-free H-induced-minor-free graphs have bounded pathwidth if and only if H belongs to a class F of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs H for which every weakly sparse H-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.
This is a joint work with É. Bonnet and R. Hickingbotham.

Xavier Goaoc, A canonical tree decomposition for order types, and some applications
2026-05-19 16:30 - 17:30
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets (which we rephrase as modules), and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.
This is joint work with Mathilde Bouvel, Valentin Féray, and Florent Koechlin.

Fernanda Rivera Omaña, Erdős-Pósa theorem for matroids
2026-05-26 16:30 - 17:30
We will look at an analogue theorem of the classical Erdős-Pósa Theorem. We prove a $GF(q)$-representable matroid analogue of Robertson and Seymour’s theorem that planar graphs have an Erdős-Pósa property. Given a matroid $N$, we prove that for every matroid $M$ with bounded branch width, $M$ either contains $r$ skew copies of $N$, or there is a small perturbation of $M$ that doesn’t contain $N$ as a minor.
This is joint work with James Davies and Meike Hatzel.

Maria Chudnovsky, Induced minors and treewidth
2026-06-02 16:30 - 17:30
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.

J. Pascal Gollin, Dominated balanced separators in wheel-induced-minor-free graphs
2026-06-09 16:30 - 17:30
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.

Harry Richman, Distinguishing graphs with tropical Weierstrass weights
2026-06-16 16:30 - 17:30
I will introduce a new structure on finite graphs, which takes the form of a labeling of the vertices by nonnegative integers (possibly repeated). This labeling is isomorphism invariant, and seems to reflect some mix of local and global structure of the graph. I will describe an algorithm for computing these labels, which uses a form of breadth-first search, and some results on the resulting labels. This construction comes from a tropical analogue of Weierstrass weights on algebraic curves, studied in the 1800s. Finally, I will discuss some speculation on whether this may or may not be helpful for the graph isomorphism problem.
This is a joint work with Omid Amini and Lucas Gierczak.

Stefan Weltge, The relaxation complexity of the standard simplex is logarithmic
2026-06-19 16:30 - 17:30
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.

Ting-Wei Chao, The Oddtown Problem Modulo a Composite Number
2026-07-10 16:30 - 17:30
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 $omega n-(2omega+varepsilon)log_2 n$ for most $n,ell$.
