## October 2023

### Matija Bucić, Essentially tight bounds for rainbow cycles in proper edge-colourings

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

### Robert Hickingbotham, Powers of planar graphs, product structure, and blocking partitions

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

### James Davies, Odd distances in colourings of the plane

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.

## November 2023

### Bruce A. Reed, Some Variants of the Erdős-Sós Conjecture

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

### Seunghun Lee (이승훈), On colorings of hypergraphs embeddable in $\mathbb{R}^d$

Room B332 IBS (기초과학연구원)

Given a hypergraph $H=(V,E)$, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:V\to$ 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

### Hyunwoo Lee (이현우), Towards a high-dimensional Dirac’s theorem

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

## December 2023

### Ben Lund, Almost spanning distance trees in subsets of finite vector spaces

Room B332 IBS (기초과학연구원)

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

### Ting-Wei Chao (趙庭偉), Tight Bound on Joints Problem and Partial Shadow Problem

Room B332 IBS (기초과학연구원)

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. Yu and I proved a tight bound on this problem, which also solves a conjecture proposed

### Shengtong Zhang (张盛桐), Triangle Ramsey numbers of complete graphs

Room B332 IBS (기초과학연구원)

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

## January 2024

### Daniel McGinnis, Applications of the KKM theorem to problems in discrete geometry

Room B332 IBS (기초과학연구원)

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

기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209