November 2023

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

Jinyoung Park (박진영), Dedekind’s Problem and beyond

Room B332 IBS (기초과학연구원)

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 $^n$. 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

Matthew Kroeker, Average flat-size in complex-representable matroids

Room B332 IBS (기초과학연구원)

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

Zichao Dong, Convex polytopes in non-elongated point sets in $\mathbb{R}^d$

Room B332 IBS (기초과학연구원)

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 $\text{diam}(P) < \alpha\sqrt{n}$ (informally speaking, `non-elongated'), contains a

February 2024

Ander Lamaison, Uniform Turán density beyond 3-graphs

Room B332 IBS (기초과학연구원)

The uniform Turán density $\pi_u(F)$ of a hypergraph $F$, introduced by Erdős and Sós, is the smallest value of $d$ such that any hypergraph $H$ where all linear-sized subsets of vertices of $H$ have density greater than $d$ contains $F$ as a subgraph. Over the past few years the value of $\pi_u(F)$ was determined for

Sebastian Wiederrecht, Packing even directed circuits quarter-integrally

Room B332 IBS (기초과학연구원)

We prove the existence of a computable function $f\colon\mathbb{N}\to\mathbb{N}$ such that for every integer $k$ and every digraph $D$ either contains a collection $\mathcal{C}$ of $k$ directed cycles of even length such that no vertex of $D$ belongs to more than four cycles in $\mathcal{C}$, or there exists a set $S\subseteq V(D)$ of size at

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