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PRODID:-//Discrete Mathematics Group - ECPv6.15.20//NONSGML v1.0//EN
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METHOD:PUBLISH
X-WR-CALNAME:Discrete Mathematics Group
X-ORIGINAL-URL:https://dimag.ibs.re.kr
X-WR-CALDESC:Events for Discrete Mathematics Group
REFRESH-INTERVAL;VALUE=DURATION:PT1H
X-Robots-Tag:noindex
X-PUBLISHED-TTL:PT1H
BEGIN:VTIMEZONE
TZID:Asia/Seoul
BEGIN:STANDARD
TZOFFSETFROM:+0900
TZOFFSETTO:+0900
TZNAME:KST
DTSTART:20220101T000000
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240123T163000
DTEND;TZID=Asia/Seoul:20240123T173000
DTSTAMP:20260416T005051
CREATED:20231120T123044Z
LAST-MODIFIED:20240707T072547Z
UID:7930-1706027400-1706031000@dimag.ibs.re.kr
SUMMARY:Zichao Dong\, Convex polytopes in non-elongated point sets in $\mathbb{R}^d$
DESCRIPTION:For any finite point set $P \subset \mathbb{R}^d$\, we denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d\, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position\, satisfying $\text{diam}(P) < \alpha\sqrt[d]{n}$ (informally speaking\, `non-elongated’)\, contains a convex $c$-polytope. Valtr proved that $c_{2\, \alpha}(n) \approx \sqrt[3]{n}$\, which is asymptotically tight in the plane. We generalize the results by establishing $c_{d\, \alpha}(n) \approx n^{\frac{d-1}{d+1}}$. Along the way we generalize the definitions and analysis of convex cups and caps to higher dimensions\, which may be of independent interest. Joint work with Boris Bukh.
URL:https://dimag.ibs.re.kr/event/2024-01-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240116T163000
DTEND;TZID=Asia/Seoul:20240116T173000
DTSTAMP:20260416T005051
CREATED:20231211T010749Z
LAST-MODIFIED:20240705T155059Z
UID:8007-1705422600-1705426200@dimag.ibs.re.kr
SUMMARY:Matthew Kroeker\, Average flat-size in complex-representable matroids
DESCRIPTION:Melchior’s Inequality (1941) implies that\, in a rank-3 real-representable matroid\, the average number of points in a line is less than three. This was extended to the complex-representable matroids by Hirzebruch in 1983 with the slightly weaker bound of four. In this talk\, we discuss and sketch the proof of the recent result that\, in a rank-4 complex-representable matroid which is not the direct-sum of two lines\, the average number of points in a plane is bounded above by an absolute constant. Consequently\, the average number of points in a flat in a rank-4 complex-representable matroid is bounded above by an absolute constant. Extensions of these results to higher dimensions will also be discussed. In particular\, the following quantities are bounded in terms of k and r respectively: the average number of points in a rank-k flat in a complex-representable matroid of rank at least 2k-1\, and the average number of points in a flat in a rank-r complex-representable matroid. Our techniques rely on a theorem of Ben Lund which approximates the number of flats of a given rank. \nThis talk is based on joint work with Rutger Campbell and Jim Geelen.
URL:https://dimag.ibs.re.kr/event/2024-01-16/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240111T163000
DTEND;TZID=Asia/Seoul:20240111T173000
DTSTAMP:20260416T005051
CREATED:20231116T155919Z
LAST-MODIFIED:20240705T155117Z
UID:7919-1704990600-1704994200@dimag.ibs.re.kr
SUMMARY:Jinyoung Park (박진영)\, Dedekind's Problem and beyond
DESCRIPTION:The Dedekind’s Problem asks the number of monotone Boolean functions\, a(n)\, on n variables. Equivalently\, a(n) is the number of antichains in the n-dimensional Boolean lattice $[2]^n$. While the exact formula for the Dedekind number a(n) is still unknown\, its asymptotic formula has been well-studied. Since any subsets of a middle layer of the Boolean lattice is an antichain\, the logarithm of a(n) is trivially bounded below by the size of the middle layer. In the 1960’s\, Kleitman proved that this trivial lower bound is optimal in the logarithmic scale\, and the actual asymptotics was also proved by Korshunov in 1980’s. In this talk\, we will discuss recent developments on some variants of Dedekind’s Problem. Based on joint works with Matthew Jenssen\, Alex Malekshahian\, Michail Sarantis\, and Prasad Tetali.
URL:https://dimag.ibs.re.kr/event/2024-01-11/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240102T163000
DTEND;TZID=Asia/Seoul:20240102T173000
DTSTAMP:20260416T005051
CREATED:20230828T061831Z
LAST-MODIFIED:20240705T161245Z
UID:7552-1704213000-1704216600@dimag.ibs.re.kr
SUMMARY:Daniel McGinnis\, Applications of the KKM theorem to problems in discrete geometry
DESCRIPTION:We present the KKM theorem and a recent proof method utilizing it that has proven to be very useful for problems in discrete geometry. For example\, the method was used to show that for a planar family of convex sets with the property that every three sets are pierced by a line\, there are three lines whose union intersects each set in the family. This was previously a long-unsolved problem posed by Eckhoff. We go over a couple of examples demonstrating the method and propose a potential future research direction to push the method even further.
URL:https://dimag.ibs.re.kr/event/2024-01-02/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231219T163000
DTEND;TZID=Asia/Seoul:20231219T173000
DTSTAMP:20260416T005051
CREATED:20231015T221647Z
LAST-MODIFIED:20240707T072612Z
UID:7757-1703003400-1703007000@dimag.ibs.re.kr
SUMMARY:Shengtong Zhang (张盛桐)\, Triangle Ramsey numbers of complete graphs
DESCRIPTION:A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$\, denoted by $r_F(H)$\, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro\, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}3\] for all sufficiently large $t$.  Our proof involves a combination of results on the chromatic number of triangle-sparse graphs. \nJoint work with Jacob Fox and Jonathan Tidor.
URL:https://dimag.ibs.re.kr/event/2023-12-19/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231212T163000
DTEND;TZID=Asia/Seoul:20231212T173000
DTSTAMP:20260416T005051
CREATED:20231019T075456Z
LAST-MODIFIED:20240707T072622Z
UID:7777-1702398600-1702402200@dimag.ibs.re.kr
SUMMARY:Ting-Wei Chao (趙庭偉)\, Tight Bound on Joints Problem and Partial Shadow Problem
DESCRIPTION:Given a set of lines in $\mathbb R^d$\, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most $O(N^{3/2})$ joints in $\mathbb R^3$ via the polynomial method. \nYu and I proved a tight bound on this problem\, which also solves a conjecture proposed by Bollobás and Eccles on the partial shadow problem. It is surprising to us that the only known proof of this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.
URL:https://dimag.ibs.re.kr/event/2023-12-12/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231204T163000
DTEND;TZID=Asia/Seoul:20231204T173000
DTSTAMP:20260416T005051
CREATED:20231127T150526Z
LAST-MODIFIED:20240707T072633Z
UID:7951-1701707400-1701711000@dimag.ibs.re.kr
SUMMARY:Ben Lund\, Almost spanning distance trees in subsets of finite vector spaces
DESCRIPTION:For $d\ge 2$ and an odd prime power $q$\, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb{F}_q$. The distance between two points $(x_1\,\ldots\,x_d)$ and $(y_1\,\ldots\,y_d)$ is defined to be $\sum_{i=1}^d (x_i-y_i)^2$. An influential result of Iosevich and Rudnev is: if $E \subset \mathbb{F}_q^d$ is sufficiently large and $t \in \mathbb{F}_q$\, then there are a pair of points $x\,y \in E$ such that the distance between $x$ and $y$ is $t$. Subsequent works considered embedding graphs of distances\, rather than a single distance. I will discuss joint work with Debsoumya Chakraborti\, in which we show that every sufficiently large subset of $\mathbb{F}_q^d$ contains every nearly spanning tree of distances with bounded degree in each distance. The main novelty in this result is that the distance graphs we find are nearly as large as the set $S$ itself\, but even for smaller distance trees our work leads to quantitative improvements to previously known bounds. A key ingredient in our proof is a new colorful generalization of a classical result of Haxell on finding nearly spanning bounded-degree trees in expander graphs. This is joint work with Debsoumya Chakraborti.
URL:https://dimag.ibs.re.kr/event/2023-12-04/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231128T163000
DTEND;TZID=Asia/Seoul:20231128T173000
DTSTAMP:20260416T005051
CREATED:20231031T121453Z
LAST-MODIFIED:20240707T072649Z
UID:7831-1701189000-1701192600@dimag.ibs.re.kr
SUMMARY:Hyunwoo Lee (이현우)\, Towards a high-dimensional Dirac's theorem
DESCRIPTION:Dirac’s theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. \nWe consider another natural generalization of the perfect matchings\, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices\, it is a natural high-dimensional analogue of a perfect matching in graphs. \nWe prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6}\,$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879… + o(1))n$ contains a Steiner triple system. In fact\, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. \nWe conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac’s theorem.
URL:https://dimag.ibs.re.kr/event/2023-11-28/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231120T163000
DTEND;TZID=Asia/Seoul:20231120T173000
DTSTAMP:20260416T005051
CREATED:20231031T024622Z
LAST-MODIFIED:20240707T072908Z
UID:7825-1700497800-1700501400@dimag.ibs.re.kr
SUMMARY:Seunghun Lee (이승훈)\, On colorings of hypergraphs embeddable in $\mathbb{R}^d$
DESCRIPTION:Given a hypergraph $H=(V\,E)$\, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:V\to [m]$ such that every hyperedge of $H$ is not monochromatic. The (weak) chromatic number of $H$\, denoted by $\chi(H)$\, is the smallest $m$ such that $H$ is $m$-colorable. A vertex subset $T \subseteq V$ is called a transversal of $H$ if for every hyperedge $e$ of $H$ we have $T\cap e \ne \emptyset$. The transversal number of $H$\, denoted by $\tau(H)$\, is the smallest size of a transversal in $H$. The transversal ratio of $H$ is the quantity $\tau(H)/|V|$ which is between 0 and 1. Since a lower bound on the transversal ratio of $H$ gives a lower bound on $\chi(H)$\, these two quantities are closely related to each other. \nUpon my previous presentation\, which is based on the joint work with Joseph Briggs and Michael Gene Dobbins (https://www.youtube.com/watch?v=WLY-8smtlGQ)\, we update what is discovered in the meantime about transversals and colororings of geometric hypergraphs. In particular\, we focus on chromatic numbers of $k$-uniform hypergraphs which are embeddable in $\mathbb{R}^d$ by varying $k$\, $d$\, and the notion of embeddability and present lower bound constructions. This result can also be regarded as an improvement upon the research program initiated by Heise\, Panagiotou\, Pikhurko\, and Taraz\, and the program by Lutz and Möller. We also present how this result is related to the previous results and open problems regarding transversal ratios. This presentation is based on the joint work with Eran Nevo.
URL:https://dimag.ibs.re.kr/event/2023-11-20/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231107T163000
DTEND;TZID=Asia/Seoul:20231107T173000
DTSTAMP:20260416T005051
CREATED:20230803T141247Z
LAST-MODIFIED:20240705T161256Z
UID:7449-1699374600-1699378200@dimag.ibs.re.kr
SUMMARY:Bruce A. Reed\, Some Variants of the Erdős-Sós Conjecture
DESCRIPTION:Determining the density required to ensure that a host graph G contains some target graph as a subgraph or minor is a natural and well-studied question in extremal combinatorics. The celebrated 50-year-old Erdős-Sós conjecture states that for every k\, if G has average degree exceeding k-2 then it contains every tree T with k vertices as a subgraph. This is tight as the clique with k-1 vertices contains no tree with k vertices as a subgraph. \nWe present some variants of this conjecture. We first consider replacing bounds on the average degree by bounds on the minimum and maximum degrees. We then consider replacing subgraph by minor in the statement.
URL:https://dimag.ibs.re.kr/event/2023-11-07/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;VALUE=DATE:20231101
DTEND;VALUE=DATE:20231106
DTSTAMP:20260416T005051
CREATED:20230911T044018Z
LAST-MODIFIED:20240705T161224Z
UID:7610-1698796800-1699228799@dimag.ibs.re.kr
SUMMARY:The 3rd East Asia Workshop on Extremal and Structural Graph Theory
DESCRIPTION:The 3rd East Asia Workshop on Extremal and Structural Graph Theory is a workshop to bring active researchers in the field of extremal and structural graph theory\, especially in the East Asia such as China\, Japan\, and Korea. \nWebsite: http://tgt.ynu.ac.jp/2023EastAsia.html
URL:https://dimag.ibs.re.kr/event/20231101/
LOCATION:The Southern Beach Hotel & Resort Okinawa
CATEGORIES:Workshops and Conferences
ORGANIZER;CN="Seog-Jin Kim (%EA%B9%80%EC%84%9D%EC%A7%84)":MAILTO:skim12@konkuk.ac.kr
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231031T163000
DTEND;TZID=Asia/Seoul:20231031T173000
DTSTAMP:20260416T005051
CREATED:20231009T124517Z
LAST-MODIFIED:20240705T161008Z
UID:7739-1698769800-1698773400@dimag.ibs.re.kr
SUMMARY:James Davies\, Odd distances in colourings of the plane
DESCRIPTION:We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd integral distance from each other.
URL:https://dimag.ibs.re.kr/event/2023-10-31/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231024T163000
DTEND;TZID=Asia/Seoul:20231024T173000
DTSTAMP:20260416T005051
CREATED:20231005T065005Z
LAST-MODIFIED:20240705T161015Z
UID:7714-1698165000-1698168600@dimag.ibs.re.kr
SUMMARY:Robert Hickingbotham\, Powers of planar graphs\, product structure\, and blocking partitions
DESCRIPTION:Graph product structure theory describes complex graphs in terms of products of simpler graphs. In this talk\, I will introduce this subject and talk about some of my recent results in this area. The focus of my talk will be on a new tool in graph product structure theory called `blocking partitions.’ I’ll show how this tool can be used to prove stronger product structure theorems for powers of planar graphs as well as $k$-planar graphs\, resolving open problems of Dujmović\, Morin and Wood\, and Ossona de Mendez.
URL:https://dimag.ibs.re.kr/event/2023-10-24/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231017T163000
DTEND;TZID=Asia/Seoul:20231017T173000
DTSTAMP:20260416T005051
CREATED:20230918T023401Z
LAST-MODIFIED:20240707T072941Z
UID:7670-1697560200-1697563800@dimag.ibs.re.kr
SUMMARY:Matija Bucić\, Essentially tight bounds for rainbow cycles in proper edge-colourings
DESCRIPTION:An edge-coloured graph is said to be rainbow if it uses no colour more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash\, Mubayi\, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on n vertices without a rainbow cycle. Improving upon a series of earlier bounds\, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently\, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon’s bound. We show that the answer to the question is equal to $(\log n)^{1+o(1)}$.\nA key tool we use is the theory of robust sublinear expanders. In addition\, we observe a connection between this problem and several questions in additive number theory\, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups.\nJoint work with: Noga Alon\, Lisa Sauermann\, Dmitrii Zakharov and Or Zamir.
URL:https://dimag.ibs.re.kr/event/2023-10-17/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;VALUE=DATE:20231015
DTEND;VALUE=DATE:20231021
DTSTAMP:20260416T005051
CREATED:20230911T044555Z
LAST-MODIFIED:20240705T161223Z
UID:7616-1697328000-1697846399@dimag.ibs.re.kr
SUMMARY:2023 Vertex-Minor Workshop
DESCRIPTION:This workshop aims to foster collaborative discussions and explore the various aspects of vertex-minors\, including structural theory and their applications. \nThis event will be small-scale\, allowing for focused talks and meaningful interactions among participants. \nWebsite: https://indico.ibs.re.kr/event/596/
URL:https://dimag.ibs.re.kr/event/2023-vertex-minor-workshop/
LOCATION:SONO BELLE Jeju
CATEGORIES:Workshops and Conferences
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20231010T163000
DTEND;TZID=Asia/Seoul:20231010T173000
DTSTAMP:20260416T005051
CREATED:20230917T133424Z
LAST-MODIFIED:20240705T161024Z
UID:7665-1696955400-1696959000@dimag.ibs.re.kr
SUMMARY:Domagoj Bradač\, Effective bounds for induced size-Ramsey numbers of cycles
DESCRIPTION:The k-color induced size-Ramsey number of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges\, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995\, in their seminal paper\, Haxell\, Kohayakawa and Łuczak showed that for cycles these numbers are linear for any constant number of colours\, i.e.\, for some C=C(k)\, there is a graph with at most Cn edges whose any k-edge-coloring contains a monochromatic induced cycle of length n. The value of C comes from the use of the sparse regularity lemma and has a tower-type dependence on k. In this work\, we obtain nearly optimal bounds for the required value of C. Joint work with Nemanja Draganić and Benny Sudakov.
URL:https://dimag.ibs.re.kr/event/2023-10-10/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230926T163000
DTEND;TZID=Asia/Seoul:20230926T173000
DTSTAMP:20260416T005051
CREATED:20230718T083922Z
LAST-MODIFIED:20240705T162115Z
UID:7392-1695745800-1695749400@dimag.ibs.re.kr
SUMMARY:Carl R. Yerger\, Solving Problems in Graph Pebbling using Optimization and Structural Techniques
DESCRIPTION:Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebbling on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that\, given any initial configuration of pebbles\, at least one pebble can be moved to a specified target vertex. \nIn this talk\, we will give a survey of several streams of research in pebbling\, including describing a theoretical and computational framework that uses mixed-integer linear programming to obtain bounds for the pebbling numbers of graphs. We will also discuss improvements to this framework through the use of newly proved weight functions that strengthen the weight function technique of Hurlbert. Finally\, we will discuss some open extremal problems in pebbling\, specifically related to Class 0 graphs and describe how structural graph theoretic techniques such as discharging can be used to obtain results. \nCollaborators on these projects include Dan Cranson\, Dominic Flocco\, Luke Postle\, Jonad Pulaj\, Chenxiao Xue\, Marshall Yang\, Daniel Zhou.
URL:https://dimag.ibs.re.kr/event/2023-09-26/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230919T163000
DTEND;TZID=Asia/Seoul:20230919T173000
DTSTAMP:20260416T005051
CREATED:20230904T063838Z
LAST-MODIFIED:20240707T073002Z
UID:7590-1695141000-1695144600@dimag.ibs.re.kr
SUMMARY:Donggyu Kim (김동규)\, Orthogonal matroids over tracts
DESCRIPTION:Even delta-matroids generalize matroids\, as they are defined by a certain basis exchange axiom weaker than that of matroids. One natural example of even delta-matroids comes from a skew-symmetric matrix over a given field $K$\, and we say such an even delta-matroid is representable over the field $K$. Interestingly\, a matroid is representable over $K$ in the usual manner if and only if it is representable over $K$ in the sense of even delta-matroids. The representability of matroids got much interest and was extensively studied such as excluded minors and representability over more than one field. Recently\, Baker and Bowler (2019) integrated the notions of $K$-representable matroids\, oriented matroids\, and valuated matroids using tracts that are commutative ring-like structures in which the sum of two elements may output no element or two or more elements. \nWe generalize Baker-Bowler’s theory of matroids with coefficients in tracts to orthogonal matroids that are equivalent to even delta-matroids. We define orthogonal matroids with coefficients in tracts in terms of Wick functions\, orthogonal signatures\, circuit sets\, and orthogonal vector sets\, and establish basic properties on functoriality\, duality\, and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular\, we give a new proof that an orthogonal matroid is regular (i.e.\, representable over all fields) if and only if it is representable over $\mathbb{F}_2$ and $\mathbb{F}_3$\, which was originally shown by Geelen (1996)\, and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over $\mathbb{F}_3$ and $\mathbb{F}_4$. \nThis is joint work with Tong Jin.
URL:https://dimag.ibs.re.kr/event/2023-09-19/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230912T163000
DTEND;TZID=Asia/Seoul:20230912T173000
DTSTAMP:20260416T005051
CREATED:20230817T015106Z
LAST-MODIFIED:20240707T073045Z
UID:7512-1694536200-1694539800@dimag.ibs.re.kr
SUMMARY:Seog-Jin Kim (김석진)\, The square of every subcubic planar graph of girth at least 6 is 7-choosable
DESCRIPTION:The square of a graph $G$\, denoted $G^2$\, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Wegner’s conjecture (1977) states that for a planar graph $G$\, the chromatic number $\chi(G^2)$ of $G^2$ is at most 7 if $\Delta(G) = 3$\, at most $\Delta(G)+5$ if $4 \leq \Delta(G) \leq 7$\, and at most $\lfloor \frac{3 \Delta(G)}{2} \rfloor$ if $\Delta(G) \geq 8$. Wegner’s conjecture is still wide open. The only case for which we know tight bound is when $\Delta(G) = 3$. Thomassen (2018) showed that $\chi(G^2) \leq 7$ if $G$ is a planar graph with $\Delta(G) = 3$\, which implies that Wegner’s conjecture is true for $\Delta(G) = 3$. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph\, where $\chi_{\ell}(G^2)$ is the list chromatic number of $G^2$. Cranston and Kim (2008) showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. This is joint work with Xiaopan Lian (Nankai University).
URL:https://dimag.ibs.re.kr/event/2023-09-12/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230905T163000
DTEND;TZID=Asia/Seoul:20230905T173000
DTSTAMP:20260416T005051
CREATED:20230823T023519Z
LAST-MODIFIED:20240707T073059Z
UID:7531-1693931400-1693935000@dimag.ibs.re.kr
SUMMARY:Sebastian Wiederrecht\, Delineating half-integrality of the Erdős-Pósa property for minors
DESCRIPTION:In 1986\, Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minor if and only if it is planar. In particular\, for every non-planar graph $H$ they gave examples showing that the Erdős-Pósa property does not hold for $H$. Recently\, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors. \nIn this talk\, we start the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph $H$ there exists a finite family $\mathfrak{F}_H$ of parametric graphs such that $H$ has the Erdős-Pósa property in a minor-closed graph class $\mathcal{G}$ if and only if $\mathcal{G}$ excludes a minor of each of the parametric graphs in $\mathfrak{F}_H$. We prove this conjecture for the class $\mathcal{H}$ of Kuratowski-connected shallow-vortex minors by showing that\, for every non-planar $H\in\mathcal{H}$ the family $\mathfrak{F}_H$ can be chosen to be precisely the two families of Robertson-Seymour counterexamples to the Erdős-Pósa property of $H$.
URL:https://dimag.ibs.re.kr/event/2023-09-05/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230829T163000
DTEND;TZID=Asia/Seoul:20230829T173000
DTSTAMP:20260416T005051
CREATED:20230706T112113Z
LAST-MODIFIED:20240707T073327Z
UID:7355-1693326600-1693330200@dimag.ibs.re.kr
SUMMARY:Dabeen Lee (이다빈)\, From coordinate subspaces over finite fields to ideal multipartite uniform clutters
DESCRIPTION:Take a prime power $q$\, an integer $n\geq 2$\, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$\, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper\, we determine when the clutter $\mathcal{C}$ is ideal\, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly\, the characterization differs depending on whether $q$ is $2\,4$\, a higher power of $2$\, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified\, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence\, we prove the Replication and $\tau=2$ conjectures for this class of clutters. This is joint work with Ahmad Abdi (London School of Economics).
URL:https://dimag.ibs.re.kr/event/2023-08-29/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230822T163000
DTEND;TZID=Asia/Seoul:20230822T173000
DTSTAMP:20260416T005051
CREATED:20230425T022914Z
LAST-MODIFIED:20240707T073336Z
UID:7057-1692721800-1692725400@dimag.ibs.re.kr
SUMMARY:Linda Cook\, Orientations of $P_4$ bind the dichromatic number
DESCRIPTION:An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. \nThe Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs\, which states that for any forest $F$\, there is some function $f$ such that every $F$-free graph $G$ with clique number $\omega(G)$ has chromatic number at most $f(\omega(G))$. \nAboulker\, Charbit\, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker\, Charbit\, and Naserasr’s $\overrightarrow{\chi}$-boundedness conjecture states that for every oriented forest $F$\, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(\omega(D))$\, where $\omega(D)$ is the size of a maximum clique in the graph underlying $D$. \nIn this talk\, we perform the first step towards proving Aboulker\, Charbit\, and Naserasr’s $\overrightarrow{\chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.
URL:https://dimag.ibs.re.kr/event/2023-08-22/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230809T163000
DTEND;TZID=Asia/Seoul:20230809T173000
DTSTAMP:20260416T005051
CREATED:20230403T234729Z
LAST-MODIFIED:20240705T164149Z
UID:6963-1691598600-1691602200@dimag.ibs.re.kr
SUMMARY:R. Amzi Jeffs\, Intersection patterns of convex sets
DESCRIPTION:How can one arrange a collection of convex sets in d-dimensional Euclidean space? This guiding question is fundamental in discrete geometry\, and can be made concrete in a variety of ways\, for example the study of hyperplane arrangements\, embeddability of simplicial complexes\, Helly-type theorems\, and more. This talk will focus on the classical topic of d-representable complexes and its more recent generalization to convex codes. We will show how these frameworks differ\, share some novel Helly-type results\, and offer several tantalizing open questions.
URL:https://dimag.ibs.re.kr/event/2023-08-09/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230809T100000
DTEND;TZID=Asia/Seoul:20230809T160000
DTSTAMP:20260416T005051
CREATED:20230802T144334Z
LAST-MODIFIED:20240707T073343Z
UID:7441-1691575200-1691596800@dimag.ibs.re.kr
SUMMARY:2023 Mini-Workshop on Discrete Geometry
DESCRIPTION:2023 Mini-Workshop on Discrete Geometry will be held on August 9th at Room B332\, Institute for Basic Science (IBS)\, Daejeon\, Republic of Korea. \nThe workshop consists of three presentations on recent results and an open problem session. \nResearchers who are highly interested in this field are welcome to attend. \nTentative schedule\n\n10:00-10:50 Michael Dobbins (SUNY Binghamton): Colorful intersections\, Tverberg partitions\, and geometric transversals\n11:00-11:30 Andreas Holmsen (KAIST): The topology of the complex of ordered partitions\n11:30-13:30 Lunch and free discussion\n13:30-14:10 Minki Kim김민기 (GIST): Some variants of the colorful Helly theorems\n14:20-16:00 Open problem session\n16:30-17:30 Discrete Math Seminar: R. Amzi Jeffs (Carnegie Mellon University)\n\nOrganizers\n\nAndreas Holmsen (KAIST)\nJinha Kim김진하 (IBS Discrete Mathematics Group)\nMinki Kim김민기 (GIST)\nSang-il Oum엄상일 (IBS Discrete Mathematics Group / KAIST)\n\nAbstracts\nMichael Dobbins (SUNY Binghamton): Colorful intersections\, Tverberg partitions\, and geometric transversals\nGiven three red convex sets and three blue convex sets in three-dimensional space\, suppose every red set intersects every blue set. Montejano’s theorem says there is a line that intersects all the red sets or all the blue sets. This was generalized to k-transversals in $\mathbb R^d$ by Montejano and Karasev using sophisticated algebraic and topological tools. Here we give further generalizations based on more accessible methods such as the test-map/configuration space scheme\, Sarkaria’s tensor method\, and discrete Morse theory. \nAndreas Holmsen (KAIST): The topology of the complex of ordered partitions\nThe set of partitions of {1\,…\,n} into k nonempty ordered parts can be equipped with a natural cell-complex structure which we denote by P(n\,k). Here we use discrete Morse theory to show that P(n\,k) is homotopy equivalent to a wedge of (n-k)-dimensional spheres\, where the number of spheres is given in terms of Stirling numbers of the second kind. Our result has applications related to geometric transversals and Tverberg partitions. \nMinki Kim (GIST): Some variants of the colorful Helly theorems\n\nGiven a finite family $F$ of nonempty sets\, the nerve of $F$ is the simplicial complex on $F$ whose simplices are precisely the intersecting subfamilies of $F$. The colorful Helly theorem\, which generalizes Helly’s theorem\, asserts the following: if $X$ is the nerve of the disjoint union of $d+1$ many finite families $F_1\,\ldots\,F_{d+1}$ of convex sets in $\mathbb{R}^d$ where each $F_i$ is not a simplex in $X$\, then there is a colorful $(d+1)$-tuple that is not a simplex of $X$. It was shown by Kalai and Meshulam that the same statement holds for $d$-collapsible/Leray complexes. In this talk\, I will explain the notion of $d$-collapsibility and $d$-Lerayness of simplicial complexes\, and present recent results on variants of the colorful Helly theorem and applications. This is based on joint work with Alan Lew.
URL:https://dimag.ibs.re.kr/event/2023-discrete-geometry/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Workshops and Conferences
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230802T163000
DTEND;TZID=Asia/Seoul:20230802T173000
DTSTAMP:20260416T005051
CREATED:20230506T225557Z
LAST-MODIFIED:20240705T163040Z
UID:7156-1690993800-1690997400@dimag.ibs.re.kr
SUMMARY:Daniel Kráľ\, High chromatic common graphs
DESCRIPTION:Ramsey’s Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. As probabilistic constructions often provide good bounds on quantities in extremal combinatorics\, we say that a graph H is common if the random 2-edge-coloring asymptotically minimizes the number of monochromatic copies of H. This notion goes back to the work of Erdős in the 1960s\, who conjectured that every complete graph is common. The conjecture was disproved by Thomason in the 1980s\, however\, a classification of common graphs remains one of the most intriguing problems in extremal combinatorics. \nSidorenko’s Conjecture (if true) would imply that every bipartite graph is common\, and in fact\, no bipartite common graph unsettled for Sidorenko’s Conjecture is known. Until Hatami et al. showed that a 5-wheel is common about a decade ago\, all graphs known to be common had chromatic number at most three. The existence of a common graph with chromatic number five or more has remained open for three decades. \nWe will present a construction of (connected) common graphs with arbitrarily large chromatic number. At the end of the talk\, we will also briefly discuss the extension of the notion to more colors and particularly its relation to Sidorenko’s Conjecture. \nThe main result presented in the talk is based on joint work with Jan Volec and Fan Wei.
URL:https://dimag.ibs.re.kr/event/2023-08-02/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230725T163000
DTEND;TZID=Asia/Seoul:20230725T173000
DTSTAMP:20260416T005051
CREATED:20230615T122924Z
LAST-MODIFIED:20240707T073405Z
UID:7282-1690302600-1690306200@dimag.ibs.re.kr
SUMMARY:Dong Yeap Kang (강동엽)\, Hamilton cycles and optimal matchings in a random subgraph of uniform Dirac hypergraphs
DESCRIPTION:A loose cycle is a cyclic ordering of edges such that every two consecutive edges share exactly one vertex. A cycle is Hamilton if it spans all vertices. A codegree of a $k$-uniform hypergraph is the minimum nonnegative integer $t$ such that every subset of vertices of size $k-1$ is contained in $t$ distinct edges. \nWe prove “robust” versions of Dirac-type theorems for Hamilton cycles and optimal matchings. \nLet $\mathcal{H}$ be a $k$-uniform hypergraph on $n$ vertices with $n \in (k-1)\mathbb{N}$ and codegree at least $n/(2k-2)$\, and let $\mathcal{H}_p$ be a spanning subgraph of $\mathcal{H}$ such that each edge of $\mathcal{H}$ is included in $\mathcal{H}_p$ with probability $p$ independently at random. We prove that a.a.s. $\mathcal{H}_p$ contains a loose Hamilton cycle if $p = \Omega(n^{-k+1} \log n)$\, which is asymptotically best possible. We also present similar results for Hamilton $\ell$-cycles for $\ell \geq 2$. \nFurthermore\, we prove that if $\mathcal{H}$ is a $k$-uniform hypergraph on $n$ vertices with $n \notin k \mathbb{N}$ and codegree at least $\lfloor n/k \rfloor$\, then a.a.s. $\mathcal{H}_p$ contains a matching of size $\lfloor n/k \rfloor$ if $p = \Omega(n^{-k+1} \log n)$. This is also asymptotically best possible. \nThis is joint work with Michael Anastos\, Debsoumya Chakraborti\, Abhishek Methuku\, and Vincent Pfenninger.
URL:https://dimag.ibs.re.kr/event/2023-07-25/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230718T163000
DTEND;TZID=Asia/Seoul:20230718T173000
DTSTAMP:20260416T005051
CREATED:20230601T142456Z
LAST-MODIFIED:20240705T163017Z
UID:7226-1689697800-1689701400@dimag.ibs.re.kr
SUMMARY:Andrzej Grzesik\, Rainbow Turán problems
DESCRIPTION:In a rainbow variant of the Turán problem\, we consider $k$ graphs on the same set of vertices and want to determine the smallest possible number of edges in each graph\, which guarantees the existence of a copy of a given graph $F$ containing at most one edge from each graph. In other words\, we treat each of the $k$ graphs as a graph in one of the $k$ colors and consider how many edges in each color force a rainbow copy of a given graph $F$. In the talk\, we will describe known results on the topic\, as well as present recent developments\, obtained jointly with Sebastian Babiński and Magdalen Prorok\, solving the rainbow Turán problem for a path on 4 vertices and a directed triangle with any number of colors.
URL:https://dimag.ibs.re.kr/event/2023-07-18/
LOCATION:Room S221\, IBS (기초과학연구원) Science Culture Center
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230713T094000
DTEND;TZID=Asia/Seoul:20230713T105000
DTSTAMP:20260416T005051
CREATED:20230711T000049Z
LAST-MODIFIED:20240705T162121Z
UID:7377-1689241200-1689245400@dimag.ibs.re.kr
SUMMARY:Open Symposium at the Discrete Mathematics Group
DESCRIPTION:Program\n\n9:40-10:05: Sang-il OUM\, Obstructions for dense analogs of tree-depth\n10:05-10:20: Kevin HENDREY\, Structural and extremal results for twin-width\n10:20-10:35: Rutger CAMPBELL\, Down-sets in combinatorial posets\n10:35-10:50: Linda COOK\, Reuniting 𝜒-boundedness with polynomial 𝜒-boundedness
URL:https://dimag.ibs.re.kr/event/2023-07-11/
LOCATION:Room B109\, IBS (기초과학연구원)
CATEGORIES:Workshops and Conferences
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230710T163000
DTEND;TZID=Asia/Seoul:20230710T173000
DTSTAMP:20260416T005051
CREATED:20230410T004309Z
LAST-MODIFIED:20240707T073602Z
UID:6987-1689006600-1689010200@dimag.ibs.re.kr
SUMMARY:Xuding Zhu (朱緒鼎)\, List version of 1-2-3 conjecture
DESCRIPTION:The well-known 1-2-3 Conjecture by Karoński\, Łuczak and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1\, 2 or 3 so that for each edge $uv$ the sums of the weights at $u$ and at $v$ are distinct. The list version of the 1-2-3 Conjecture by Bartnicki\, Grytczuk and Niwczyk states that the same holds if each edge $e$ has the choice of weights not necessarily from $\{1\,2\,3\}$\, but from any set $\{x(e)\,y(e)\,z(e)\}$ of three real numbers. The goal of this talk is to survey developments on the 1-2-3 Conjecture\, especially on the list version of the 1-2-3 Conjecture.
URL:https://dimag.ibs.re.kr/event/2023-07-10/
LOCATION:Room B109\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20230704T163000
DTEND;TZID=Asia/Seoul:20230704T173000
DTSTAMP:20260416T005051
CREATED:20230627T045021Z
LAST-MODIFIED:20240705T162129Z
UID:7316-1688488200-1688491800@dimag.ibs.re.kr
SUMMARY:Tuan Tran\, Complexity of null dynamical systems
DESCRIPTION:A theoretical dynamical system is a pair (X\,T) where X is a compact metric space and T is a self homeomorphism of X. The topological entropy of a theoretical dynamical system (X\,T)\, first introduced in 1965 by Adler\, Konheim and McAndrew\, is a nonnegative real number that measures the complexity of the system. Systems with positive entropy are random in certain sense\, and systems with zero entropy are said to be deterministic. To distinguish between deterministic systems\, Huang and Ye (2009) introduced the concept of maximal pattern entropy of a theoretical dynamical system. At the heart of their argument is a Sauer-Shelah-type lemma. We will discuss this lemma and its surprising connection to a recent breakthrough in communication complexity. \nJoint work with Guorong Gao\, Jie Ma\, and Mingyuan Rong.
URL:https://dimag.ibs.re.kr/event/2023-07-04/
LOCATION:Room B109\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
END:VEVENT
END:VCALENDAR