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.
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The 3rd East Asia Workshop on Extremal and Structural Graph Theory
The Southern Beach Hotel & Resort OkinawaThe 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. Website: http://tgt.ynu.ac.jp/2023EastAsia.html
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 …
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 …
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 …