Discrete Mathematics Group

Vadim Lozin, Graph problems and monotone classes
2024-05-21 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
Very little is known about critical properties of graphs in the hierarchy of monotone classes, i.e. classes closed under taking (not necessarily induced) subgraphs. We distinguish four important levels in this hierarchy and discuss possible new levels by focusing on the Hamiltonian cycle problem. In particular, we obtain a number of results for this problem on monotone classes.

Yongho Shin (신용호), Three-way online correlated selection
2024-05-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm’s choices. OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020, JACM 2022) to break a natural long-standing barrier in edge-weighted online bipartite matching. They posed two open questions, one of which was the following: Can we obtain n-way OCS for $n >2$, in which the algorithm can be given $n >2$ elements to choose from at each timestep?
In this talk, we affirmatively answer this open question by presenting a three-way OCS which is simple to describe: it internally runs two instances of two-way OCS, one of which is fed with the output of the other. Contrast to its simple construction, we face a new challenge in analysis that the final output probability distribution of our three-way OCS is highly elusive since it requires the actual output distribution of two-way OCS. We show how we tackle this challenge by approximating the output distribution of two-way OCS by a flatter distribution serving as a safe surrogate.
This is joint work with Hyung-Chan An.

Jane Tan, Semi-strong colourings of hypergraphs
2024-06-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $min{c, |e|}$ distinct colours. Let $chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $chi(t,c)$ is finite: they showed that $chi(t,c)$ is finite whenever $t geq c$ and unbounded when $tleq c-2$. The boundary case $chi(c-1, c)$ has remained elusive for some time: $chi(1,2)$ is known to be finite by an easy classical result, and $chi(2,3)$ was shown to be finite by Chung and independently by Colucci and Gyárfás in 2013. In this talk, we present some recent work with Kevin Hendrey, Freddie Illingworth and Nina Kamčev in which we fill in this gap by showing that $chi(c-1, c)$ is finite in general.

Maria Chudnovsky, Anticomplete subgraphs of large treewidth
2024-06-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We will discuss recent progress on the topic of induced subgraphs and tree-decompositions. In particular this talk with focus on the proof of a conjecture of Hajebi that asserts that (if we exclude a few obvious counterexamples) for every integer t, every graph with large enough treewidth contains two anticomplete induced subgraphs each of treewidth at least t. This is joint work with Sepher Hajebi and Sophie Spirkl.

Semin Yoo (유세민), Paley-like quasi-random graphs arising from polynomials
2024-06-18 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs defined by polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs associated with Diophantine tuples and their generalizations from number theory. As an application, we provide new lower bounds on the clique number and independence number of general quasi-random graphs. In particular, we give a sufficient condition for the clique number of quasi-random graphs of order $n$ to be at least $(1-o(1))log_{3.008}n$. Such a condition applies to many classical quasi-random graphs, including Paley graphs and Paley sum graphs, as well as some new Paley-like graphs we construct. If time permits, we also discuss some problems of diophantine tuples arising from number theory, which is our original motivation.
This is joint work with Seoyoung Kim and Chi Hoi Yip.

Wonwoo Kang (강원우), Skein relations for punctured surfaces
2024-06-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Since the introduction of cluster algebras by Fomin and Zelevinsky in 2002, there has been significant interest in cluster algebras of surface type. These algebras are particularly noteworthy due to their ability to construct various combinatorial structures like snake graphs, T-paths, and posets, which are useful for proving key structural properties such as positivity or the existence of bases. In this talk, we will begin by presenting a cluster expansion formula that integrates the work of Musiker, Schiffler, and Williams with contributions from Wilson, utilizing poset representatives for arcs on triangulated surfaces. Using these posets and the expansion formula as tools, we will demonstrate skein relations, which resolve intersections or incompatibilities between arcs. Topologically, a skein relation replaces intersecting arcs or arcs with self-intersections with two sets of arcs that avoid the intersection differently. Additionally, we will show that all skein relations on punctured surfaces include a term that is not divisible by any coefficient variable. Consequently, we will see that the bangles and bracelets form spanning sets and exhibit linear independence. This work is done in collaboration with Esther Banaian and Elizabeth Kelley.

Kisun Lee (이기선), Symmetric Tropical Rank 2 Matrices
2024-07-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. We discuss that this tropicalization has a simplicial complex structure as the space of symmetric bicolored trees. As a result, we show that this space is shellable and delve into its matroidal structure. It is based on the joint work with May Cai and Josephine Yu.

Hyunwoo Lee (이현우), Random matchings in linear hypergraphs
2024-07-05 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a given hypergraph $H$ and a vertex $vin V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $vin V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998. In this paper, we disprove this conjecture for all $k geq 3.$ For infinitely many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $mathcal{P}(v_1
otin M) = 1 – frac{(1 + o_d(1))}{d^{k-2}}$ and $mathcal{P}(v_2
otin M) = frac{(1 + o_d(1))}{d+1}.$ The gap between $mathcal{P}(v_1
otin M)$ and $mathcal{P}(v_2
otin M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

IBS-DIMAG workshop on combinatorics and geometric measure theory
2024-07-15 - 2024-07-19 | Room B109, IBS (기초과학연구원)
Website: https://cgmt.dimag.kr/
Arrival Date: July 14, 2024 Sunday.
Departure Date: July 20, 2024 Saturday.
Organizers

Ben Lund (IBS Discrete Mathematics Group)
Doowon Koh (Chungbuk National University)
Sang-il Oum (IBS Discrete Mathematics Group / KAIST)

2024 Summer School on Combinatorics and Algorithms (2024 조합론 및 알고리즘 여름학교)
2024-07-22 - 2024-07-26 | Bldg. N1, KAIST
The 2024 Summer School on Combinatorics and Algorithms is a venue for students and early-career researchers to learn selected topics in theoretical computer science and discrete mathematics. It will be a great opportunity for young and aspiring researchers to study topics which are important but not covered during the lectures in the university classes. This summer, two lecture series, combinatorial optimization and grid minor theorem, will be given by two leading experts on the subjects. There will be exercise sessions where you form a team and solve challenging questions related to the lecture subjects.
Website: https://combialgo.dimag.kr/
Lecturers and Topics

Chien-Chung Huang (ENS Paris, France): Combinatorial Optimization

This lecture (12.5h) will cover essential topics in combinatorial optimization including: Berge’s theorem, Konig’s theorem, Egervary’s theorem, Karger’s min-cut algorithm and Gomory-Hu trees, Edmonds’ blossom algorithm for maximum matching, matroid 101, multi-commodity flow and k-coverage problems.

Sebastian Wiederrecht (DIMAG-IBS, Korea): From treewidth to grid minor theorem

This lecture (6h) will present the notion of tree decomposition, treewidth and graph minor, and introduce the grid minor theorem by Robertson and Seymour. Grid minor theory is deemed as one of the most important theory in modern graph theory and has many applications in algorithms design, data structure, logic, etc.

2024 Korean Student Combinatorics Workshop (KSCW2024, 2024 조합론 학생 워크샵)
2024-07-29 - 2024-08-02 | Gongju Hanok Village, Gongju
Venue
Gongju Hanok Vilage (공주한옥마을)
Organizers

Donggyu Kim (김동규), KAIST and IBS Discrete Mathematics Group
Seokbeom Kim (김석범), KAIST and IBS Discrete Mathematics Group
Seonghyuk Im (임성혁), KAIST and IBS Extremal Combinatorics and Probability Group
Hyunwoo Lee (이현우), KAIST and IBS Extremal Combinatorics and Probability Group

Euiwoong Lee (이의웅), Parameterized Approximability of F-Deletion Problems
2024-07-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a family F of graphs, the F-Deletion Problem asks to remove the minimum number of vertices from a given graph G to ensure that G belongs to F. One of the most common ways to obtain an interesting family F is to fix another family H of graphs and let F be the set of graphs that do not contain any graph H as some notion of a subgraph, including (standard) subgraph, induced subgraph, and minor. This framework captures numerous basic graph problems, including Vertex Cover, Feedback Vertex Set, and Treewidth Deletion, and provides an interesting forum where ideas from approximation and parameterized algorithms influence each other. In this talk, I will give a brief survey on the state of the art on the F-Deletion Problems for the above three notions of subgraphs, and talk about a recent result on Weighted Bond Deletion.

Daniel Král', Matroid depth and width parameters
2024-08-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Depth and width parameters of graphs, e.g., tree-width, path-width and tree-depth, play a crucial role in algorithmic and structural graph theory. These notions are of fundamental importance in the theory of graph minors, fixed parameter complexity and the theory of sparsity.
In this talk, we will survey structural and algorithmic results that concern width and depth parameters of matroids. We will particularly focus on matroid depth parameters and discuss the relation of the presented concepts to discrete optimization. As an application, we will present matroid based algorithms that uncover a hidden Dantzig-Wolfe-like structure of an input instance (if such structure is present) and transform instances of integer programming to equivalent ones, which are amenable to the existing tools in integer programming.
The most recent results presented in the talk are based on joint work with Marcin Briański, Jacob Cooper, Timothy F. N. Chan, Martin Koutecký, Ander Lamaison, Kristýna Pekárková and Felix Schröder.

Peter Nelson, Formalizing matroid theory in a proof assistant
2024-08-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For the past few years, I’ve been working on formalizing proofs in matroid theory using the Lean proof assistant. This has led me to many interesting and unexpected places. I’ll talk about what formalization looks like in practice from the perspective of a combinatorialist.

2024 Workshop on (Mostly) Matroids
2024-08-19 - 2024-08-23 | IBS Science Culture Center
The 2024 Workshop on (Mostly) Matroids will be held in-person at the Institute for Basic Science (IBS), Daejeon, South Korea, from August 19, 2024 to August 23, 2024. We expect that most people would arrive on Sunday, August 18 and leave on Saturday, August 24.
Our hope is that this workshop will continue the tradition of previous workshops held in Sittard (2008), Maastricht (2010,2012), Princeton (2014), Eindhoven(2016), Waterloo (2017), and Baton Rouge (2019). The focus will be on all aspects of matroid theory, including its connection to graph theory, algebraic geometry, and other areas of mathematics.

Dillon Mayhew, Monadic second-order definability for gain-graphic matroids
2024-08-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Every (finite) matroid consists of a (finite) set called the ground set, and a collection of distinguished subsets called the independent sets. A classic example arises when the ground set is a finite set of vectors from a vector space, and the independent subsets are exactly the subsets that are linearly independent. Any such matroid is said to be representable. We can think of a representable matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic.
Monadic second-order logic is a natural language for matroid applications. In this language we are able to quantify only over subsets of the ground set. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle’s Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects). It is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder.
This talk will contain a brief introduction to matroids. Based on work with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Angus Matthews, Mike Newman, and Gabriel Verret.

2024 Combinatorics Workshop (2024 조합론 학술대회)
2024-08-28 - 2024-08-30 | Chungbuk National University
Website: https://cw2024.combinatorics.kr/
Location
Chungbuk National University, Cheongju, Korea.
Advisory Committee

Committee of Discrete Mathematics, The Korean Mathematical Society (Chair: Sang-il Oum, IBS Discrete Mathematics Group / KAIST)

Sponsors

IBS Discrete Mathematics Group.
Korean Mathematical Society

Amadeus Reinald, Oriented trees in $O(k sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture
2024-09-03 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $frac{k^2}{2} – frac{k}{2} + 1$ by Addario-Berry et al.
In this talk, we give the first subquadratic bound for Burr’s conjecture, by showing that every directed graph with chromatic number $8sqrt{frac{2}{15}} k sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $sqrt{frac{4}{3}} k sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $bge 2$.

Neal Bushaw, Edge-colored Extremal Problems
2024-09-06 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An edge-colored graph $H$ is called rainbow if all of its edges are given distinct colors. An edge-colored graph $G$ is then called rainbow $H$-free when no copy of $H$ in $G$ is rainbow. With this, we define a graph $G$ to be rainbow $H$-saturated when there is some proper edge-coloring of $G$ which is rainbow $H$-free, but for every pair of non-adjacent vertices $x,yin V(G)$, the graph $G+xy$ formed by adding the edge $xy$ to $G$ cannot be given a rainbow $H$-free coloring. We think of these graphs as edge-maximal rainbow $H$-free graphs. (We note that here we make no restrictions on the colorings of $G+xy$ whatsoever, except that they are proper colorings. They may use any number of colors, and need not be extensions of the original rainbow $H$-free coloring of $G$.)

With this framework in place, we define the rainbow saturation number and rainbow extremal number to be the largest and smallest number of edges, respectively, among all $n$ vertex rainbow $H$-saturated graphs. The latter of these was defined by Keevash, Mubayi, Sudakov, and Verstraëte in 2007; the former was introduced by B., Johnston, and Rombach in 2019.

In this talk, we discuss recent progress on both the rainbow saturation numbers and rainbow extremal numbers. We also give several broad generalizations of these concepts and discuss many open problems. This talk contains joint work with Vic Bednar (Furman), Dan Johnston (Trinity College, CT), and Puck Rombach (Vermont).

Zhihan Jin (金之涵), The Helly number of Hamming balls and related problems
2024-09-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove the following variant of Helly’s classical theorem for Hamming balls with a bounded radius. For $n > t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X| > 1$ and all $n > t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory. This is joint work with N. Alon and B. Sudakov.

Gábor Tardos, Extremal theory of 0-1 matrices
2024-09-24 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We say that a 0-1 matrix A contains another such matrix (pattern) P if P can be obtained from a submatrix of A by possibly changing a few 1 entries to 0. The main question of this theory is to estimate the maximal number of 1 entries in an n by n 0-1 matrix NOT containing a given pattern P. This question has very close connections to Turan type extremal graph theory and also to the Devenport-Schinzel theory of sequences. Results in the extremal theory of 0-1 matrices proved useful in attacking problems in far away fields as combinatorial geometry and the analysis of algorithms.
This talk will concentrate on acyclic patterns and survey some old and recent results in the area and will also contain several open problems.

Mathias Schacht, Canonical colourings in random graphs
2024-10-08 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $ellgeq 2$ they showed that $n^{-frac{2}{ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_ell$.
We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erdős and Rado showed that canonically coloured copies of $K_ell$ can be ensured in the deterministic setting.
We transfer the Erdős-Rado theorem to the random environment and show that for $ellgeq 4$ both thresholds coincide. As a consequence the proof yields $K_{ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_ell$.
This is joint work with Nina Kamčev.

Kyeongsik Nam (남경식), Random walks on percolation
2024-10-15 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In general, random walks on fractal graphs are expected to exhibit anomalous behaviors, for example heat kernel is significantly different from that in the case of lattices. Alexander and Orbach in 1982 conjectured that random walks on critical percolation, a prominent example of fractal graphs, exhibit mean field behavior; for instance, its spectral dimension is 4/3. In this talk, I will talk about this conjecture for a canonical dependent percolation model.

Colin Geniet, Permutations, patterns, and twin-width
2024-10-22 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
This talk will first introduce combinatorics on permutations and patterns, presenting the basic notions and some fundamental results: the Marcus-Tardos theorem which bounds the density of matrices avoiding a given pattern, and the Guillemot-Marx algorithm for pattern detection using the notion now known as twin-width.
I will then present a decomposition result: permutations avoiding a pattern factor into bounded products of separable permutations. This can be rephrased in terms of twin-width: permutation with bounded twin-width are build from a bounded product of permutations of twin-width 0. Comparable results on graph encodings follow from this factorisation.
This is joint work with Édouard Bonnet, Romain Bourneuf, and Stéphan Thomassé.

Felix Christian Clemen, Triangles in the Plane
2024-10-29 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A classical problem in combinatorial geometry, posed by Erdős in 1946, asks to determine the maximum number of unit segments in a set of $n$ points in the plane. Since then a great variety of extremal problems in finite planar point sets have been studied. Here, we look at such questions concerning triangles. Among others we answer the following question asked by Erdős and Purdy almost 50 years ago: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles?
For our proofs we use hypergraph Turán theory. This is joint work with Balogh and Dumitrescu.

Michał Pilipczuk, Monadic stability and monadic dependence
2024-11-05 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We will give an overview of the recent attempts of building a structure theory for graphs centered around First-Order transductions: a notion of containment inspired by finite model theory. Particularly, we will speak about the notions of monadic dependence and monadic stability, their combinatorial characterizations, and the developments on the algorithmic front.

IBS-DIMAG Workshop on Topology and Combinatorics
2024-11-11 11:00 - 17:30 | Room B332, IBS (기초과학연구원)
The IBS-DIMAG Workshop on Topology and Combinatorics will be held on November 11, 2024 at Room B332, Institute for Basic Science (IBS), Daejeon, South Korea.
Invited Speakers (tentative)

Karim Adiprasito (Jussieu Institute of Mathematics)
Minho Cho조민호 (IBS Extremal Combinatorics and Probability Group)
Niloufar Fuladi (INRIA Center of Université de Lorraine)
Minki Kim김민기 (GIST)
Dohyeon Lee이도현 (KAIST & IBS Discrete Mathematics Group)
Geunho Lim임근호 (Jussieu Institute of Mathematics)
Semin Yoo유세민 (IBS Discrete Mathematics Group)

Tentative Schedule

11:00-11:30 Karim Adiprasito, The Charney Davis conjecture and Boolean decompositions
12:00-14:00 Lunch Break
14:00-14:30 Niloufar Fuladi, Universal families of arcs and curves on surfaces
14:30-15:00 Semin Yoo유세민, The Charney-Davis conjecture and its related complexes (survey)
15:00-15:30 Geunho Lim임근호, Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
15:30-16:00 Break
16:00-16:30 Minki Kim김민기, Some extensions of the colorful Helly theorem
16:30-17:00 Minho Cho조민호, Fractional Helly theorems via weak saturation
17:00-17:30 Dohyeon Lee이도현, Betti number of clique complex of H-free graphs (survey)
18:00-20:00 Banquet

Organizer

Semin Yoo유세민, IBS Discrete Mathematics Group

Abstracts
Karim Adiprasito, The Charney Davis conjecture and Boolean decompositions
I will present a novel approach to the Charney Davis conjecture by introducing anticommutative Lefschetz properties, and studying them on the Danzer complex of flag spheres.
Semin Yoo, The Charney-Davis conjecture and its related complexes (survey)
The Charney-Davis conjecture lies in the intersection between topology and combinatorics. The conjecture is a special case of the Hopf-Chern-Thurston conjecture for all nonpositively curved, piecewise Euclidean manifolds which are cellulated by regular Euclidean cubes. In 2001, Davis and Okun solved the conjecture for a 4-dimensional case, and some other partial results are also known. However, the conjecture is still open so far.
In this talk, I will describe five statements that solve the Charney-Davis conjecture and explain some complexes to achieve this goal.
Niloufar Fuladi, Universal families of arcs and curves on surfaces
In this talk I introduce a family of curves that realize all pants decomposition of a surface: a family of simple closed curves Γ on a surface realizes all types of pants decompositions if for any pants decomposition of the surface, there exists a homeomorphism sending it to a subset of the curves in Γ. The study of such universal families of curves is motivated by questions on graph embeddings, joint crossing numbers and finding an elusive center of moduli space. I will discuss the case of surfaces without punctures, where we establish an exponential upper bound and a superlinear lower bound on the minimum size of the family of curves with this universal property. I also talk about a similar concept of universality for triangulations of polygons, where we provide bounds which are tight up to logarithmic factors. In this talk, I will explain the background and context and no prior knowledge in topology or surfaces is needed.
This is joint work with Arnaud de Mesmay and Hugo Parlier.
Geunho Lim, Bounds on Cheeger-Gromov invariants and simplicial complexity of triangulated manifolds
Using L^2 cohomology, Cheeger and Gromov define the L^2 rho-invariant on manifolds with arbitrary fundamental groups, as a generalization of the Atiyah-Singer rho-invariant. There are many interesting applications in geometry, topology, and combinatorics. In this talk, we show linear bounds on the rho-invariants in terms of simplicial complexity of manifolds by using hyperbolization methods. As applications, we give new concrete examples in the complexity theory of high-dimensional (homotopy) lens spaces. This is a joint work with Shmuel Weinberger.
Minki Kim, Some extensions of the colorful Helly theorem
Given $k > d$ finite families of convex sets in $mathbb{R}^d$, if every colorful $(d+1)$-tuple is intersecting, then the union of $k-d$ color classes is intersecting. This is known as the colorful Helly theorem. Based on recent joint work with Alan Lew, I will present some extensions of the colorful Helly theorem and their applications.
Minho Cho, Fractional Helly theorems via weak saturation
Two celebrated extensions of the classical Helly’s theorem are the fractional Helly theorem and the colorful Helly theorem. Bulavka, Goodarzi, and Tancer recently established the optimal bound for the unified generalization of the fractional and the colorful Helly theorems using a colored extension of the exterior algebra. In this talk, we introduce a combinatorial reduction of both the fractional Helly theorem and its colorful version to a classical problem in extremal combinatorics known as weak saturation. No such results connecting the fractional Helly theorem and weak saturation are known in the long history of literature. These reductions, along with basic linear algebraic arguments for the reduced weak saturation problems, let us give new short proofs of the optimal bounds for both the fractional Helly theorem and its colorful version without using exterior algebra.
This is joint work with Debsoumya Chakraborti, Jinha Kim, and Minki Kim.
Dohyeon Lee, Betti number of clique complex of H-free graphs (survey)
The clique complex of a graph is a simplicial complex whose faces correspond to cliques of the given graph. In this talk, we investigate extremal properties of the Betti numbers of clique complexes. While general simplicial complexes with n vertices can have a total Betti number approximately $2^n$, Adamaszek showed that Betti number of clique complex is bounded by around $1.32^n$. Furthermore, if the underlying graph has an independence number at most two, the bound tightens to $1.25^n$. Adiprasito, Nevo, and Tancer established additional bounds on the total Betti number for clique complexes of (induced) H-free graphs for various graph H, along with constructions of graphs with large total Betti number. In another direction, Zhang and Wu demonstrated that for ternary graphs (whose with no induced cycles of length 3n), the Betti number of the independence complex (the clique complex of the complement of a graph) is at most one. Indeed, Jinha Kim showed that these independence complexes are either contractible or homotopy spheres.

Karim Adiprasito, Ehrhart theory revisited: Algebraic aspects, unimodality and more
2024-11-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Ehrhart theory is the study of lattice polytopes, specifically aimed at understanding how many lattice points are inside dilates of a given lattice polytope, and the study has a wide range of connections ranging from coloring graphs to mirror symmetry and representation theory. Recently, we introduced new algebraic tools to understand this theory, and resolve some classical conjectures. I will explain the combinatorial underpinnings behind two of the key techniques: Parseval identities for semigroup algebras, and the character algebra of a semigroup.

Eng Keat Hng, Graphon branching processes and fractional isomorphism
2024-11-26 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In 2005, Bollobás, Janson and Riordan introduced and extensively studied a general model of inhomogeneous random graphs parametrised by graphons. In particular, they studied the emergence of a giant component in these inhomogeneous random graphs by relating them to a broad collection of inhomogeneous Galton-Watson branching processes.
Fractional isomorphism of finite graphs is an important and well-studied concept at the intersection of graph theory and combinatorial optimisation. It has many different characterizations that involve a range of very different and seemingly unrelated properties of graphs. Recently, Grebík and Rocha developed a theory of fractional isomorphism for graphons.
In our work, we characterise inhomogeneous random graphs that yield the same inhomogeneous Galton-Watson branching process (and hence have a similar component structure).
This is joint work with Jan Hladký and Anna Margarethe Limbach.

Yulai Ma, Pairwise disjoint perfect matchings in regular graphs
2024-12-03 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An $r$-graph is an $r$-regular graph in which every odd set of vertices is connected to its complement by at least $r$ edges. A central question regarding $r$-graphs is determining the maximum number of pairwise disjoint perfect matchings they can contain. This talk explores how edge connectivity influences this parameter.
For ${0 leq lambda leq r}$, let $m(lambda,r)$ denote the maximum number $s$ such that every $lambda$-edge-connected $r$-graph contains $s$ pairwise disjoint perfect matchings. The values of $m(lambda,r)$ are known only in limited cases; for example, $m(3,3)=m(4,r)=1$, and $m(r,r) leq r-2$ for all $r
ot = 5$, with $m(r,r) leq r-3$ when $r$ is a multiple of $4$. In this talk, we present new upper bounds for $m(lambda,r)$ and examine connections between $m(5,5)$ and several well-known conjectures for cubic graphs.
This is joint work with Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf.
