Rob Morris, Ramsey theory: searching for order in chaos
2023-04-27 16:15 - 17:15 (Asia/Seoul) | 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.

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.

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 $H$.

Seog-Jin Kim (김석진), The square of every subcubic planar graph of girth at least 6 is 7-choosable
2023-09-12 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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).

Donggyu Kim (김동규), Orthogonal matroids over tracts
2023-09-19 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
We 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 $mathbb{F}_4$.
This is joint work with Tong Jin.

Carl R. Yerger, Solving Problems in Graph Pebbling using Optimization and Structural Techniques
2023-09-26 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
In 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.
Collaborators on these projects include Dan Cranson, Dominic Flocco, Luke Postle, Jonad Pulaj, Chenxiao Xue, Marshall Yang, Daniel Zhou.

Domagoj Bradač, Effective bounds for induced size-Ramsey numbers of cycles
2023-10-10 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.

Matija Bucić, Essentially tight bounds for rainbow cycles in proper edge-colourings
2023-10-17 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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 $(log n)^{1+o(1)}$.
A 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.
Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.

Robert Hickingbotham, Powers of planar graphs, product structure, and blocking partitions
2023-10-24 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.

James Davies, Odd distances in colourings of the plane
2023-10-31 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd integral distance from each other.

Bruce A. Reed, Some Variants of the Erdős-Sós Conjecture
2023-11-07 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
We 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.

Seunghun Lee (이승훈), On colorings of hypergraphs embeddable in $mathbb{R}^d$
2023-11-20 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
Given a hypergraph $H=(V,E)$, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:Vto [m]$ 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 $Tcap e
e emptyset$. 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.
Upon 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.

Hyunwoo Lee (이현우), Towards a high-dimensional Dirac's theorem
2023-11-28 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
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.
We 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.
We 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.
We 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.

Ben Lund, Almost spanning distance trees in subsets of finite vector spaces
2023-12-04 16:30 - 17:30 | Room B332, IBS (기초과학연구원)
For $dge 2$ 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.
