Discrete Mathematics Group

Seokbeom Kim (김석범), The structure of △(1, 2, 2)-free tournaments
2025-05-13 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
Given a tournament $S$, a tournament is $S$-free if it has no subtournament isomorphic to $S$. Until now, there have been only a small number of tournaments $S$ such that the complete structure of $S$-free tournaments is known.
Let $triangle(1, 2, 2)$ be a tournament obtained from the cyclic triangle by substituting two-vertex tournaments for two of its vertices. In this talk, we present a structure theorem for $triangle(1, 2, 2)$-free tournaments, which was previously unknown. As an application, we provide tight bounds for the chromatic number as well as the size of the largest transitive subtournament for such tournaments.
This talk is based on joint work with Taite LaGrange, Mathieu Rundström, Arpan Sadhukhan, and Sophie Spirkl.

Meike Hatzel, Counterexample to Babai's lonely colour conjecture
2025-05-27 16:30 - 17:30
Motivated by colouring minimal Cayley graphs, in 1978 Babai conjectured that no-lonely-colour graphs have bounded chromatic number. We disprove this in a strong sense by constructing graphs of arbitrarily large girth and chromatic number that have a proper edge colouring in which each cycle contains no colour exactly once.
The result presented is the joint work with James Davies and Liana Yepremyan.

Denys Bulavka, Strict Erdős-Ko-Rado Theorems for Simplicial Complexes
2025-06-04 16:30 - 17:30
The now classical theorem of Erdős, Ko and Rado establishes
the size of a maximal uniform family of pairwise-intersecting sets as well as a characterization of the families attaining such upper bound. One natural extension of this theorem is that of restricting the possiblechoices for the sets. That is, given a simplicial complex, what is the size of a maximal uniform family of pairwise-intersecting faces. Holroyd and Talbot, and Borg conjectured that the same phenomena as in the classical case (i.e., the simplex) occurs: there is a maximal size pairwise-intersecting family with all its faces having some common vertex. Under stronger hypothesis, they also conjectured that if a family attains such bound then its members must have a common vertex. In this talk I will present some progress towards the characterization of the maximal families. Concretely I will show that the conjecture is true for near-cones of sufficiently high depth. In particular, this implies that the characterization of maximal families holds for, for example, the independence complex of a chordal graph with an isolated vertex as well as the independence complex of a (large enough) disjoint union of graphs with at least one isolated vertex. Under stronger hypothesis, i.e., more isolated vertices, we also recover a stability theorem.
This talk is based on a joint work with Russ Woodroofe.

On-Hei Solomon Lo, Minors of non-hamiltonian graphs
2025-06-10 16:30 - 17:30
A seminal result of Tutte asserts that every 4-connected planar graph is hamiltonian. By Wagner’s theorem, Tutte’s result can be restated as: every 4-connected graph with no $K_{3,3}$ minor is hamiltonian. In 2018, Ding and Marshall posed the problem of characterizing the minor-minimal 3-connected non-hamiltonian graphs. They conjectured that every 3-connected non-hamiltonian graph contains a minor of $K_{3,4}$, $mathfrak{Q}^+$, or the Herschel graph, where $mathfrak{Q}^+$ is obtained from the cube by adding a new vertex and connecting it to three vertices that share a common neighbor in the cube. We recently resolved this conjecture along with some related problems. In this talk, we review the background and discuss the proof.

Attila Jung, The Quantitative Fractional Helly Theorem
2025-06-17 16:30 - 17:30
Two celebrated extensions of Helly’s theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of Barany, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family $F$ of $n$ convex sets in $mathbb{R}^d$ such that at least $alpha binom{n}{d+1}$ of the $(d+1)$-tuples of $F$ have an intersection of volume at least 1, then one can select $Omega_{d,alpha}(n)$ members of $F$ whose intersection has volume at least $Omega_d(1)$. Joint work with Nora Frankl and Istvan Tomon.

Roohani Sharma, Uniform and Constructive Polynomial Kernel for Deletion to $K_{2,p}$ Minor-Free Graphs
2025-06-25 16:30 - 17:30
Let $mathcal F$ be a fixed, finite family of graphs. In the $mathcal F$-Deletion problem, the input is a graph G and a positive integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of $mathcal F$ as a minor. The $mathcal F$-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-$eta$ Deletion, Treedepth-$eta$ Deletion, Pathwidth-$eta$ Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the $mathcal F$-Deletion problem from the kernelization perspective.
In a seminal work, Fomin, Lokshtanov, Misra & Saurabh [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family $mathcal F$: that is, the size of the kernel is $k^{f(mathcal F)}$, for some function f that depends only on $mathcal F$. Also the size of the kernel depends on non-constructive constants.
Later Giannopoulou, Jansen, Lokshtanov & Saurabh [TALG 2017] showed that the non-uniformity in the kernel size bound of Fomin et al. is unavoidable as Treewidth-$eta$ Deletion, cannot admit a kernel of size $O(k^{(eta +1)/2 -epsilon)}$, for any $epsilon >0$, unless NP $subseteq$ coNP/poly. On the other hand it was also shown that Treedepth-$eta$ Deletion, admits a uniform kernel of size $f(mathcal F) k^6$, showcasing that there are subclasses of $mathcal F$ where the asymptotic kernel sizes do not grow as a function of the family $mathcal F$. This work leads to the natural question of determining classes of $mathcal F$ where the problem admits uniform polynomial kernels.
In this work, we show that if all the graphs in $mathcal F$ are connected and $mathcal F$ contains $K_{2,p}$ (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size $f(mathcal F) k^{10}$ where the constants in the size bound are also constructive. The graph $K_{2,p}$ is one natural extension of the graph $theta_p$, where $theta_p$ is a graph on two vertices and p parallel edges. The case when $mathcal F$ contains $theta_p$ has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel.
This is joint work with William Lochet.

Sergey Norin, Asymptotic dimension of intersection graphs
2025-07-01 16:30 - 17:30
The notion of asymptotic dimension of metric spaces, introduced by Gromov, describes their large-scale behaviour. Asymptotic dimension of graph families has been recently studied, in particular, by Bonamy et al. who proved that the asymptotic dimension of proper minor-closed graph families is at most two.
We will discuss nerve-type theorems for asymptotic dimension. In particular, we show that the asymptotic dimension of intersection graphs of balls and spheres in $mathbb{R}^d$ is at most $d+1$.
Based on joint work with Zdeněk Dvořák and with Chun-Hung Liu.

Mihyun Kang (강미현), Phase transitions in a random subgraph of the hypercube
2025-07-08 16:30 - 17:30
We will discuss classical and recent results about phase transitions in random subgraphs of the hypercube and beyond. The focus will be on the giant component, long cycles, large matchings, and isoperimetric properties.

Linda Cook, A tight algorithmic meta-theorem for distributed certification within bounded treewidth graphs
2025-07-22 16:30 - 17:30
A local certification of a graph property is a protocol in which nodes are given “certificates of a graph property” that allow the nodes to check whether their network has this property while only communicating with their local network. The key property of a local certification is that if certificates are corrupted, some node in the network will be able to recognize this. Inspired by practical concerns, the aim in LOCAL certification is to minimize the maximum size of a certificate.
In this talk we introduce local certification and open problems in the area and present some recent joint work with Eunjung Kim and Tomáš Masařík, A Tight Meta-theorem for LOCAL Certification of MSO2 Properties within Bounded Treewidth Graphs.
In this work, instead of considering a specific graph property and developing a local certification protocol tailor-made for this property, we aim for generic protocols that can certify any property expressible in a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $Omega(n^2/log n)$ on general $n$-vertex graphs (Göös, Suomela 2016). Hence, we impose additional structural restrictions on the graph. Inspired by their importance in centralized computing and Robertson-Seymour Graph Minor theory, we consider graphs of bounded treewidth. We provide a local certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a local certification protocol for certifying bounded treewidth. That is, for each integer $k$ and each MSO$_2$-expressible property $Pi$, we give a local certification protocol to certify that a graph satisfies $Pi$ and has treewidth at most $k$ using certificates of size $mathcal{O}(log n)$ (which is asymptotically optimal). Our result improves upon the works of Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang (PODC 2025).

Colin Geniet, Merge-width
2025-07-29 16:30 - 17:30
This talk is an introduction to the recent notion of merge-width, proposed by Jan Dreier and Szymon Torúnczyk. I will give an overview of the context and motivations for merge-width, namely the first-order model checking problem, and present the definition, some examples, and some basic proof techniques with the example of χ-boundedness.
This is based on joint work with Marthe Bonamy.

Tony Huynh, Rainbow triangles and the Erdős-Hajnal problem in projective geometries
2025-08-05 16:30 - 17:30
We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs. In fact, we give a natural extension of the ‘multicoloured’ version of the Erdős-Hajnal conjecture. Roughly, our conjecture states that every colouring of the points of a finite projective geometry of dimension $n$ not containing a fixed colouring of a fixed projective geometry $H$ must contain a subspace of dimension polynomial in $n$ avoiding some colour.
When $H$ is a ‘triangle’, there are three different colourings, all of which we resolve. We handle the case that $H$ is a ‘rainbow’ triangle by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. The two non-rainbow colourings of $H$ are handled via a recent breakthrough result in additive combinatorics due to Kelley and Meka.
This is joint work with Carolyn Chun, James Dylan Douthitt, Wayne Ge, Matthew E. Kroeker, and Peter Nelson.

Chien-Chung Huang, Robust Sparsification for Matroid Intersection with Applications
2025-08-12 16:30 - 17:30
The matroid intersection problem is a fundamental problem in combinatorial optimization. In this problem we are given two matroids and the goal is to find the largest common independent set in both matroids. This problem was introduced and solved by Edmonds in the 70s. The importance of matroid intersection stems from the large variety of combinatorial optimization problems it captures; well-known examples in computer science include bipartite matching and packing of spanning trees/arborescences.
In this talk, we introduce a “sparsifer” for the matroid intersection problem and use it to design algorithms for two problems closely related to streaming: a one-way communication protocol and a streaming algorithm in the random-order streaming model.
This is a joint-work with François Sellier.

Sang-il Oum (엄상일), The Erdős-Pósa property for circle graphs as vertex-minors
2025-08-29 16:30 - 17:30
We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$. This is joint work with Rutger Campbell, J. Pascal Gollin, Meike Hatzel, O-joung Kwon, Rose McCarty, and Sebastian Wiederrecht.

Zhifei Yan, A Rainbow version of Lehel's conjecture
2025-09-02 16:30 - 17:30
Lehel’s conjecture states that every 2-edge-colouring of the complete graph $K_n$ admits a partition of its vertices into two monochromatic cycles. This was proven for sufficiently large n by Luczak, Rödl, and Szemerédi (1998), extended by Allen (2008), and fully resolved by Bessy and Thomassé in 2010.
We consider a rainbow version of Lehel’s conjecture for properly edge-coloured complete graphs. We prove that for any properly edge-coloured $K_n$ with sufficiently large n, there exists a partition of the vertex set into two rainbow cycles, each containing no two edges of the same colour.
This is joint work with Pedro Araújo, Xiaochuan Liu, Taísa Martins, Walner Mendonça, Luiz Moreira, and Vinicius Fernandes dos Santos.

Katherine Perry, Symmetry breaking in trees
2025-09-09 16:30 - 17:30
We will discuss two symmetry breaking parameters: distinguishing number and fixing number. Despite being introduced independently, they share meaningful connections. In particular, we show that if a tree is 2-distinguishable with order at least 3, it suffices to fix at most 4/11 of the vertices and if a tree is $d$-distinguishable, $d geq 3$, it suffices to fix at most $frac{d-1}{d+1}$ of the vertices. We also characterize the $d$-distinguishable trees with radius $r$, for any $d geq 2$ and $r geq 1$.
This is joint work with Calum Buchanan, Peter Dankleman, Isabel Harris, Paul Horn, and Emily Rivett-Carnac.

Mujin Choi (최무진), Excluding ladder and wheel as induced minor in graphs without induced stars
2025-09-16 16:30 - 17:30
We prove that for all positive integers $k$ and $d$, the class of $K_{1,d}$-free graphs not containing the $k$-ladder or the $k$-wheel as an induced minor has a bounded tree-independence number. Our proof uses a generalization of the concept of brambles to tree-independence number. This is based on joint work with Claire Hilaire, Martin Milanič, and Sebastian Wiederrecht.

Rong Luo, Modulo flows and Integer flows of signed graphs
2025-09-22 16:30 - 17:30
Nowhere-zero flows of unsigned graphs were introduced by Tutte in 1954 as a dual problem to vertex-coloring of (unsigned) planar graphs. The definition of nowhere-zero flows on signed graphs naturally comes from the study of embeddings of graphs in non-orientable surfaces, where nowhere-zero flows emerge as the dual notion to local tensions. Nowhere-zero flows in signed graphs were introduced by Edmonds and Johnson in 1970 for expressing algorithms on matchings, but systematically studied first by Bouchet in 1983. Bouchet also stated a conjecture which occupies a central place in the area of signed graphs: Every flow-admissible signed graph admits a nowhere-zero 6-flow. There is a significant difference in the flows of signed graphs and unsigned graphs. In this talk I will talk about the progress on the development of the flow theory of signed graphs.

Marcelo Sales, On the Ramsey number of Daisies and other hypergraphs
2025-09-30 16:30 - 17:30
Given a $k$-uniform hypergraph $H$, the Ramsey number $R(H;q)$ is the smallest integer $N$ such that any $q$-coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$.
When $H$ is a complete hypergraph, a classical argument of Erdős, Hajnal, and Rado reduces the general problem to the case of uniformity $k = 3$. In this talk, we will survey constructions that lift Ramsey numbers to higher uniformities and discuss recent progress on quantitative bounds for $R(H;q)$ for certain families of hypergraphs.
This is joint work with Ayush Basu, Dániel Dobák, Pavel Pudlák, and Vojtěch Rödl.

Ilkyoo Choi (최일규), An improved lower bound on the number of edges in list critical graphs via DP coloring
2025-10-14 16:30 - 17:30
A graph $G$ is (list, DP) $k$-critical if the (list, DP) chromatic number is $k$ but for every proper subgraph $G’$ of $G$, the (list, DP) chromatic number of $G’$ is less than $k$. For $kgeq 4$, we show a bound on the minimum number of edges in a DP $k$-critical graph, and our bound is the first bound that is asymptotically better than the corresponding bound for proper $k$-critical graphs by Gallai from 1963. Our result also improves the best bound on the list chromatic number. This is joint work with Bradshaw, Kostochka, and Xu.

William Cook, Optimization via Branch Decomposition
2025-10-21 16:30 - 17:30
Robertson and Seymour introduced branch-width as a connectivity invariant of graphs in their proof of the Wagner conjecture. Decompositions based on this invariant provide a natural framework for implementing dynamic-programming algorithms to solve graph optimization problems. We will discuss the computational issues involved in using branch-width as as a general tool in discrete optimization.

Jakob Greilhuber, A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth
2025-10-28 16:30 - 17:30
Vertex Cover is perhaps the most-studied problem in parameterized complexity that frequently serves as a testing ground for new concepts and techniques. In this talk, I will focus on a generalization of Vertex Cover called Component Order Connectivity (COC). Given a graph G, an integer k and a positive integer d, the task is to decide whether there is a vertex set S of size at most k such that each connected component of G – S has size at most d. If d = 1, then COC is the same as Vertex Cover.
While almost all techniques to obtain polynomial kernels for Vertex Cover extend well to COC parameterized by k + d, the same cannot be said for structural parameters. Vertex Cover admits a polynomial kernel parameterized by the vertex deletion distance to treewidth 1 graphs, but not when parameterized by the deletion distance to treewidth 2 graphs. The picture changes when considering COC: It was recently shown that COC does not admit a polynomial kernel parameterized by the vertex deletion distance to treewidth 1 graphs with pathwidth 2, even if d ≥ 2 is a fixed constant.
Complementing this, we show that COC does admit a polynomial kernel parameterized by the distance to graphs with pathwidth at most 1 (plus d). Hence, the deletion distance to pathwidth 1 vs. pathwidth 2 forms a similar line of tractability for COC as the distance to treewidth 1 vs. treewidth 2 does for Vertex Cover. In this talk, I will highlight the ideas and techniques that make this kernelization result possible.

Ahmed Ghazy and Tim Hartmann, Continuous Graphs – An Overview and a Coloring Problem
2025-11-04 16:30 - 17:30
We consider a continuous model of graphs, introduced by Dearing and Francis in 1974, where each edge of G to be a unit interval, giving rise to an infinite metric space that contains not only the vertices of G but all points on all edges of G. Several standard graph problems can be defined and studied on continuous graphs, yielding many surprising algorithmic results and combinatorial connections.
The motivation can be exemplified by the well-known Independent Set problem on graphs. Given a graph G, we want to place k facilities that are pairwise at least a distance 2 edge lengths apart. In some applications, such as when the underlying graph represents a street network, it is reasonable to allow placing a facility not only at a crossroad but also somewhere within a street, that is, not only at a vertex but also at any point on an edge between two vertices. This motivates the study of the Independent Set problem on continuous graphs. In such a setting, for example, the problem corresponds to requiring pairwise distance r=2 for the placed facilities. However, we may also study the problem where we fix r to any positive integer, rational, or even irrational number. Other problems studied in the continuous model include Dominating Set, TSP, and Coloring.
In the first part of this talk, we will give a general overview of research on continuous graphs and computational problems in this setting.
In the second part, we explore, as part of recent work, a coloring problem on continuous graphs akin to the well-known Hadwiger-Nelson Problem.
Based on joint work with Fabian Frei, Florian Hörsch, Tom Janßen, Stefan Lendl, Dániel Marx, Prahlad Narasimhan, and Gerhard Woeginger.

Simón Piga, Turán problem in hypergraphs with quasirandom links
2025-11-11 16:30 - 17:30
Given a $k$-uniform hypergraph $F$, its Turán density $pi(F)$ is the infimum over all $din [0,1]$ such that any $n$-vertex $k$-uniform hypergraph $H$ with at least $dbinom{n}{k}+o(n^k)$ edges contains a copy of $F$. While Turán densities are generally well understood for graphs ($k=2$), the problem becomes notoriously difficult for $kgeq 3$, even for small hypergraphs.
We study two well-known variants of this Turán problem for hypergraphs: first, under minimum codegree conditions and, second, with a quasirandom edge distribution. Each variant defines a distinct extremal parameter, generalising the classical Turán density. Here we present recent results in both settings, with a particular emphasis on the case of hypergraphs where every link is itself quasirandom. Our results include exact solutions for key hypergraphs and general results about the behaviour of the Turán density functions.

Fedor Noskov, Polynomial dependencies in hypergraph Turan-type problems
2025-11-18 16:30 - 17:30
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.
