기초과학연구원 이산수학그룹
A family $\mathcal F$ of subsets of {1,2,…,n} is called maximal k-wise intersecting if every collection of at most k members from $\mathcal F$ has a common element, and moreover, no set can be added to $\mathcal F$ while preserving this property. In 1974, Erdős and Kleitman asked for the smallest possible size of a …
A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected 2-complex every link graph of which is 3-connected admits an essentially unique locally flat embedding into the 3-sphere, if it admits one at all. This can be thought of …
The poset Ramsey number $R(Q_{m},Q_{n})$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_{N}$ has a blue induced copy of $Q_{m}$ or a red induced copy of $Q_{n}$. Axenovich and Walzer showed that $n+2\le R(Q_{2},Q_{n})\le2n+2$. Recently, Lu and Thompson improved the upper bound to $\frac{5}{3}n+2$. In …
What is the largest number $f(d)$ where every graph with average degree at least $d$ contains a subdivision of $K_{f(d)}$? Mader asked this question in 1967 and $f(d) = \Theta(\sqrt{d})$ was proved by Bollobás and Thomason and independently by Komlós and Szemerédi. This is best possible by considering a disjoint union of $K_{d,d}$. However, this …
The independent domination number of a graph $G$, denoted $i(G)$, is the minimum size of an independent dominating set of $G$. In this talk, we prove a series of results regarding independent domination of graphs with bounded maximum degree. Let $G$ be a graph with maximum degree at most $k$ where $k \ge 1$. We prove that …
We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) certain random variable on the boundary of a ball has a small tail. As applications, we show that the volume of intersection of …
Motivated from the surrounding property of a point set in $\mathbb{R}^d$ introduced by Holmsen, Pach and Tverberg, we consider the transversal number and chromatic number of a simplicial sphere. As an attempt to give a lower bound for the maximum transversal ratio of simplicial $d$-spheres, we provide two infinite constructions. The first construction gives infinitely …
A geometric transversal to a family of convex sets in $\mathbb R^d$ is an affine flat that intersects the members of the family. While there exists a far-reaching theory concerning 0-dimensional transversals (intersection patterns of convex sets), much less is known when it comes to higher-dimensional transversals. In this talk, I will present some new …
For given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a 'colorful' copy of $H$, i.e. a copy of $H$ containing at most one edge from each $G_i$? Keevash, Saks, Sudakov, and Verstraëte defined $\operatorname{ex}_k(n,H)$ to be the maximum total number of edges of the graphs …
In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomassé, and Watrigant defined the twin-width of a graph $G$ to be the minimum integer …