기초과학연구원 이산수학그룹
Let $X$ be a real random variable; a typical anti-concentration inequality asserts that (under certain assumptions) if an interval $I$ has small length, then $\mathbb{P}(X\in I)$ is small, regardless the location of $I$. Inequalities of this type have found powerful applications in many branches of mathematics. In this talk we will discuss several recent applications …
In the early 1980s, Beck proved that, if P is a set of n points in the real plane, and no more than g points of P lie on any single line, then there are $\Omega(n(n-g))$ lines that each contain at least 2 points of P. In 2016, I found a generalization of this theorem, …
I give a quick survey on stability and NIP(Non-Independent Property). We first review basic facts on the first order logic and give some historical remarks on classification theory in model theory. We review basic properties of stability and NIP. Finally, we aim to give several characterizations of stability and NIP of a given formula in terms of …
I give a quick survey on stability and NIP(Non-Independent Property). We first review basic facts on the first order logic and give some historical remarks on classification theory in model theory. We review basic properties of stability and NIP. Finally, we aim to give several characterizations of stability and NIP of a given formula in terms of …
I give a quick survey on stability and NIP(Non-Independent Property). We first review basic facts on the first order logic and give some historical remarks on classification theory in model theory. We review basic properties of stability and NIP. Finally, we aim to give several characterizations of stability and NIP of a given formula in terms of …
Let E be a finite set and I be a collection of subsets of E. When is there a set of real vectors indexed by E such that I correspond to its linearly independent subsets? In 1935, Whitney introduced matroids using some necessary conditions for this. However, complete characterizations with various techniques are intractable. This remains the case even if it is already known …
Generalized extremal problems have been one of the central topics of study in extremal combinatorics throughout the last few decades. One such simple-looking problem, maximizing the number of cliques of a fixed order in a graph with a fixed number of vertices and given maximum degree, was recently resolved by Chase. Settling a conjecture of …
Let $G$ be a graph on the vertex set $V$. A vertex subset $W \subset V$ is a cover of $G$ if $V \setminus W$ is an independent set of $G$, and $W$ is a non-cover of $G$ if $W$ is not a cover of $G$. The non-cover complex of $G$ is a simplicial complex …
Let $G$ be a graph on $V$ and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose faces are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of $I_n(G)$ for various classes of graphs, focusing on the class of …
For any given graph $H$, one may define a natural corresponding functional $\|.\|_H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once $H$ is paired with a $2$-edge-colouring $\alpha$ to assign conjugates. We say that $H$ is real-norming (resp. complex-norming) if $\|.\|_H$ (resp. there is $\alpha$ such that …