Younjin Kim (김연진), Problems on Extremal Combinatorics
2023-03-21 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
Extremal Combinatorics studies the maximum or minimum size of finite objects (numbers, sets, graphs) satisfying certain properties. In this talk, I introduce the conjectures I solved on Extremal Combinatorics, and also introduce recent extremal problems.

Qizhong Lin, Two classical Ramsey-Turán numbers involving triangles
2023-03-22 16:30 - 17:30 | Zoom ID: 224 221 2686 (ibsecopro)
In 1993, Erdős, Hajnal, Simonovits, Sós and Szemerédi proposed to determine the value of Ramsey-Turán density $rho(3,q)$ for $qge3$. Erdős et al. (1993) and Kim, Kim and Liu (2019) proposed two corresponding conjectures. However, we only know four values of this Ramsey-Turán density by Erdős et al. (1993). There is no progress on this classical Ramsey-Turán density since then. In this talk, I will introduce two new values of this classical Ramsey-Turán density. Moreover, the corresponding asymptotically extremal structures are weakly stable, which answers a problem of Erdős et al. (1993) for the two cases. Joint work with Xinyu Hu.

Tianchi Yang, On the maximum number of edges in k-critical graphs
2023-03-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
In this talk, we will discuss the problem of determining the maximum number of edges in an n-vertex k-critical graph. A graph is considered k-critical if its chromatic number is k, but any proper subgraph has a chromatic number less than k. The problem remains open for any integer k ≥ 4. Our presentation will showcase an improvement on previous results achieved by employing a combination of extremal graph theory and structural analysis. We will introduce a key lemma, which may be of independent interest, as it sheds light on the partial structure of dense k-critical graphs. This is joint work with Cong Luo and Jie Ma.

István Tomon, Configurations of boxes
2023-04-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Configurations of axis-parallel boxes in $mathbb{R}^d$ are extensively studied in combinatorial geometry. Despite their perceived simplicity, there are many problems involving their structure that are not well understood. I will talk about a construction that shows that their structure might be more complicated than people conjectured.

Jie Han, Spanning trees in expanders
2023-04-06 10:00 - 11:00 | Zoom ID: 224 221 2686 (ibsecopro)
We consider the spanning tree embedding problem in dense graphs without bipartite holes and sparse graphs. In 2005, Alon, Krivelevich and Sudakov asked for determining the best possible spectral gap forcing an $(n,d,lambda)$-graph to be $T(n, Delta)$-universal. In this talk, we introduce our recent work on this conjecture.

James Davies, Two structural results for pivot-minors
2023-04-11 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Pivot-minors can be thought of as a dense analogue of graph minors. We shall discuss pivot-minors and two recent results for proper pivot-minor-closed classes of graphs. In particular, that for every graph H, the class of graphs containing no H-pivot-minor is 𝜒-bounded, and also satisfies the (strong) Erdős-Hajnal property.

MATRIX-IBS Workshop: Structural Graph Theory Downunder III
2023-04-17 - 2023-04-24 | MATRIX, Australia
Website: https://www.matrix-inst.org.au/events/structural-graph-theory-downunder-iii/
Program Description:
This program, jointly organised by MATRIX and the Discrete Mathematics Group of the Korean Institute for Basic Science (IBS), builds on the “Structural Graph Theory Downunder” programs held at MATRIX in 2019 and 2022. In this short intensive workshop, mathematicians from across the globe will come together to work on open problems in structural graph theory. The focus will be on graph colouring, graph minors, induced subgraphs, and graph product structure theory. The majority of the time will be allocated to collaborative research, with only a few talks describing recent advances. The goal is to create an environment where mathematicians at all career stages work side-by-side. We anticipate that open problems will be solved, and lasting collaborations will be initiated.
Confirmed Participants:
Rutger Campbell (Institute for Basic Science, Korea)
Linda Cook (Institute for Basic Science, Korea)
James Davies (University of Cambridge, UK)
Marc Distel (Monash University, Australia)
Zdeněk Dvořák (Charles University, Czech Republic)
Bryce Frederickson (Emory University, USA)
António Girão (University of Oxford, UK)
Pascal Gollin (Institute for Basic Science, Korea)
Kevin Hendrey (Institute for Basic Science, Korea)
Robert Hickingbotham (Monash University, Australia)
Freddie Illingworth (University of Oxford, UK)
O-joung Kwon (Hanyang University, Korea)
Florian Lehner (University of Auckland, New Zealand)
Anita Liebenau (UNSW, Australia)
Chun-Hung Liu (Texas A&M, USA)
Rose McCarty (Princeton University, USA)
Lukas Michel (University of Oxford, UK)
Sang-il Oum (Institute for Basic Science and KAIST, Korea)
Michael Savery (University of Oxford, UK)
Alex Scott (University of Oxford, UK)
Raphael Steiner (ETH Zurich, Switzerland)
Jane Tan (University of Oxford, UK)
Sebastian Wiederrecht (Institute for Basic Science, Korea)
David Wood (Monash University, Australia)
Liana Yepremyan (Emory University, USA)

Shin-ichiro Seki, On the extension of the Green-Tao theorem to number fields
2023-04-19 16:30 - 17:30 | Zoom ID: 897 6822 0619 (ibsecopro) [04/19 only]
In 2006, Tao established the Gaussian counterpart of the celebrated Green-Tao theorem on arithmetic progressions of primes. In this talk, I will explain the extension of Tao’s theorem and the Green-Tao theorem to the case of general number fields. Our combinatorial tool is the relative hypergraph removal lemma by Conlon-Fox-Zhao. I will discuss the difficulties that arise in the case of general number fields and an application of our results to prime representations by binary quadratic forms. This is based on joint work with Wataru Kai, Masato Mimura, Akihiro Munemasa, and Kiyoto Yoshino.

Hyunwoo Lee (이현우), On perfect subdivision tilings
2023-04-25 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For a given graph $H$, we say that a graph $G$ has a perfect $H$-subdivision tiling if $G$ contains a collection of vertex-disjoint subdivisions of $H$ covering all vertices of $G.$ Let $delta_{sub}(n, H)$ be the smallest integer $k$ such that any $n$-vertex graph $G$ with minimum degree at least $k$ has a perfect $H$-subdivision tiling. For every graph $H$, we asymptotically determined the value of $delta_{sub}(n, H)$. More precisely, for every graph $H$ with at least one edge, there is a constant $1 < xi^*(H)leq 2$ such that $delta_{sub}(n, H) = left(1 - frac{1}{xi^*(H)} + o(1) right)n$ if $H$ has a bipartite subdivision with two parts having different parities. Otherwise, the threshold depends on the parity of $n$.

Rob Morris, Ramsey theory: searching for order in chaos
2023-04-27 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
In many different areas of mathematics (such as number theory, discrete geometry, and combinatorics), one is often presented with a large “unstructured” object, and asked to find a smaller “structured” object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and present some recent progress on one of the central problems in the area: bounding the so-called “diagonal” Ramsey numbers. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

Rob Morris, An exponential improvement for diagonal Ramsey
2023-05-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Ramsey number $R(k)$ is the minimum n such that every red-blue colouring of the edges of the complete graph on n vertices contains a monochromatic copy of $K_k$. It has been known since the work of Erdős and Szekeres in 1935, and Erdős in 1947, that $2^{k/2} < R(k) < 4^k$, but in the decades since the only improvements have been by lower order terms. In this talk I will sketch the proof of a very recent result, which improves the upper bound of Erdős and Szekeres by a (small) exponential factor.
Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

Jozef Skokan, Separating the edges of a graph by a linear number of paths
2023-05-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Recently, Letzter proved that any graph of order n contains a collection P of $O(n log^*n)$ paths with the following property: for all distinct edges e and f there exists a path in P which contains e but not f. We improve this upper bound to 19n, thus answering a question of Katona and confirming a conjecture independently posed by Balogh, Csaba, Martin, and Pluhar and by Falgas-Ravry, Kittipassorn, Korandi, Letzter, and Narayanan.
Our proof is elementary and self-contained.

Maria Chudnovsky, Induced subgraphs and tree decompositions
2023-05-11 16:15 - 17:15 | Room 1501, Bldg. E6-1, KAIST
Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

Oliver Janzer, Small subgraphs with large average degree
2023-05-16 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We study the fundamental problem of finding small dense subgraphs in a given graph. For a real number $s>2$, we prove that every graph on $n$ vertices with average degree at least $d$ contains a subgraph of average degree at least $s$ on at most $nd^{-frac{s}{s-2}}(log d)^{O_s(1)}$ vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with $n$ vertices and average degree at least $n^{1-frac{2}{s}+varepsilon}$ contains a subgraph of average degree at least $s$ on $O_{varepsilon,s}(1)$ vertices, which is also optimal up to the constant hidden in the $O(.)$ notation, and resolves a conjecture of Verstraëte.
Joint work with Benny Sudakov and Istvan Tomon.

Szymon Toruńczyk, Flip-width: Cops and Robber on dense graphs
2023-05-17 16:00 - 17:00 | Zoom ID: 869 4632 6610 (ibsdimag)
We define new graph parameters, called flip-width, that generalize treewidth, degeneracy, and generalized coloring numbers for sparse graphs, and clique-width and twin-width for dense graphs. The flip-width parameters are defined using variants of the Cops and Robber game, in which the robber has speed bounded by a fixed constant r∈N∪{∞}, and the cops perform flips (or perturbations) of the considered graph. We then propose a new notion of tameness of a graph class, called bounded flip-width, which is a dense counterpart of classes of bounded expansion of Nešetril and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomassé, and Watrigant. This unifies Sparsity Theory and Twin-width Theory, providing a common language for studying the central notions of the two theories, such as weak coloring numbers and twin-width — corresponding to winning strategies of one player — or dense shallow minors, rich divisions, or well-linked sets, corresponding to winning strategies of the other player. We prove that boundedness of flip-width is preserved by first-order interpretations, or transductions, generalizing previous results concerning classes of bounded expansion and bounded twin-width. We provide an algorithm approximating the flip-width of a given graph, which runs in slicewise polynomial time (XP) in the size of the graph. Finally, we propose a more general notion of tameness, called almost bounded flip-width, which is a dense counterpart of nowhere dense classes. We conjecture, and provide evidence, that classes with almost bounded flip-width coincide with monadically dependent (or monadically NIP) classes, introduced by Shelah in model theory. We also provide evidence that classes of almost bounded flip-width characterise the hereditary graph classes for which the model-checking problem is fixed-parameter tractable.

Suyun Jiang (江素云), How connectivity affects the extremal number of trees
2023-05-30 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
The Erdős-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $frac {n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a $k$-vertex tree $T$, we construct $n$-vertex connected graphs that are $T$-free with at least $(1/4-o_k(1))nk$ edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of $k$-vertex brooms $T$ such that the maximum size of an $n$-vertex connected $T$-free graph is at most $(1/4+o_k(1))nk$.

Minho Cho (조민호), Strong Erdős-Hajnal property on chordal graphs and its variants
2023-06-13 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
A graph class $mathcal{G}$ has the strong Erdős-Hajnal property (SEH-property) if there is a constant $c=c(mathcal{G}) > 0$ such that for every member $G$ of $mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m geq leftlfloor c|V(G)| rightrfloor$. We prove that the class of chordal graphs satisfies SEH-property with constant $c = 2/9$.
On the other hand, a strengthening of SEH-property which we call the colorful Erdős-Hajnal property was discussed in geometric settings by Alon et al.(2005) and by Fox et al.(2012). Inspired by their results, we show that for every pair $F_1, F_2$ of subtree families of the same size in a tree $T$ with $k$ leaves, there exist subfamilies $F’_1 subseteq F_1$ and $F’_2 subseteq F_2$ of size $theta left( frac{ln k}{k} left| F_1 right|right)$ such that either every pair of representatives from distinct subfamilies intersect or every such pair do not intersect. Our results are asymptotically optimal.
Joint work with Andreas Holmsen, Jinha Kim and Minki Kim.

Guanghui Wang (王光辉), Embeddings in uniformly dense hypergraphs
2023-06-20 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
An archetype problem in extremal combinatorics is to study the structure of subgraphs appearing in different classes of (hyper)graphs. We will focus on such embedding problems in uniformly dense hypergraphs. In precise, we will mention the uniform Turan density of some hypergraphs.

Chong Shangguan (上官冲), The hat guessing number of graphs
2023-06-27 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Consider the following hat guessing game: $n$ players are placed on $n$ vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph $G$, its hat guessing number $HG(G)$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
In 2008, Butler, Hajiaghayi, Kleinberg, and Leighton asked whether the hat guessing number of the complete bipartite graph $K_{n,n}$ is at least some fixed positive (fractional) power of $n$. We answer this question affirmatively, showing that for sufficiently large $n$, $HG(K_{n,n})ge n^{0.5-o(1)}$. Our guessing strategy is based on some ideas from coding theory and probabilistic method.
Based on a joint work with Noga Alon, Omri Ben-Eliezer, and Itzhak Tamo.

Tuan Tran, Complexity of null dynamical systems
2023-07-04 16:30 - 17:30 | Room B109, IBS (기초과학연구원)
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.
Joint work with Guorong Gao, Jie Ma, and Mingyuan Rong.

Xuding Zhu (朱緒鼎), List version of 1-2-3 conjecture
2023-07-10 16:30 - 17:30 | Room B109, IBS (기초과학연구원)
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.

Open Symposium at the Discrete Mathematics Group
2023-07-13 09:40 - 10:50 | Room B109, IBS (기초과학연구원)
Program

9:40-10:05: Sang-il OUM, Obstructions for dense analogs of tree-depth
10:05-10:20: Kevin HENDREY, Structural and extremal results for twin-width
10:20-10:35: Rutger CAMPBELL, Down-sets in combinatorial posets
10:35-10:50: Linda COOK, Reuniting 𝜒-boundedness with polynomial 𝜒-boundedness

Andrzej Grzesik, Rainbow Turán problems
2023-07-18 16:30 - 17:30 | Room S221, IBS (기초과학연구원) Science Culture Center
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.

Dong Yeap Kang (강동엽), Hamilton cycles and optimal matchings in a random subgraph of uniform Dirac hypergraphs
2023-07-25 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
We prove “robust” versions of Dirac-type theorems for Hamilton cycles and optimal matchings.
Let $mathcal{H}$ 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 $ell geq 2$.
Furthermore, we prove that if $mathcal{H}$ is a $k$-uniform hypergraph on $n$ vertices with $n
otin k mathbb{N}$ 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.
This is joint work with Michael Anastos, Debsoumya Chakraborti, Abhishek Methuku, and Vincent Pfenninger.

Daniel Kráľ, High chromatic common graphs
2023-08-02 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
Sidorenko’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.
We 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.
The main result presented in the talk is based on joint work with Jan Volec and Fan Wei.

2023 Mini-Workshop on Discrete Geometry
2023-08-09 10:00 - 16:00 | Room B332, IBS (기초과학연구원)
2023 Mini-Workshop on Discrete Geometry will be held on August 9th at Room B332, Institute for Basic Science (IBS), Daejeon, Republic of Korea.
The workshop consists of three presentations on recent results and an open problem session.
Researchers who are highly interested in this field are welcome to attend.
Tentative schedule

10:00-10:50 Michael Dobbins (SUNY Binghamton): Colorful intersections, Tverberg partitions, and geometric transversals
11:00-11:30 Andreas Holmsen (KAIST): The topology of the complex of ordered partitions
11:30-13:30 Lunch and free discussion
13:30-14:10 Minki Kim김민기 (GIST): Some variants of the colorful Helly theorems
14:20-16:00 Open problem session
16:30-17:30 Discrete Math Seminar: R. Amzi Jeffs (Carnegie Mellon University)

Organizers

Andreas Holmsen (KAIST)
Jinha Kim김진하 (IBS Discrete Mathematics Group)
Minki Kim김민기 (GIST)
Sang-il Oum엄상일 (IBS Discrete Mathematics Group / KAIST)

Abstracts
Michael Dobbins (SUNY Binghamton): Colorful intersections, Tverberg partitions, and geometric transversals
Given 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.
Andreas Holmsen (KAIST): The topology of the complex of ordered partitions
The 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.
Minki Kim (GIST): Some variants of the colorful Helly theorems

Given 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.

R. Amzi Jeffs, Intersection patterns of convex sets
2023-08-09 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.

Linda Cook, Orientations of $P_4$ bind the dichromatic number
2023-08-22 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
The 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 $f(omega(G))$.
Aboulker, 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 $D$.
In 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.

Dabeen Lee (이다빈), From coordinate subspaces over finite fields to ideal multipartite uniform clutters
2023-08-29 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Take a prime power $q$, an integer $ngeq 2$, and a coordinate subspace $Ssubseteq GF(q)^n$ 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).

Sebastian Wiederrecht, Delineating half-integrality of the Erdős-Pósa property for minors
2023-09-05 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
In 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 $Hinmathcal{H}$ the family $mathfrak{F}_H$ can be chosen to be precisely the two families of Robertson-Seymour counterexamples to the Erdős-Pósa property of $H$.
