기초과학연구원 이산수학그룹
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 …
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 …
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 …
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 …
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 …
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 …
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 …
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 …
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 $c_{d, \alpha}(n)$ …
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 …