기초과학연구원 이산수학그룹
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 …
Forbidden graph characterizations provide a convenient way of specifying graph classes, which often exhibit a rich combinatorial and algorithmic theory. A prime example in graph theory are classes of bounded tree-width, which are characterized as those classes that exclude some planar graph as a minor. Similarly, in model theory, classes of structures are characterized by …
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 …
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and I showed that every graph with no $K_t$ minor is …
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 …
We study the Excluded Grid Theorem, a fundamental structural result in graph theory, that was proved by Robertson and Seymour in their seminal work on graph minors. The theorem states that there is a function $f$, such that for every integer $g > 0$, every graph of treewidth at least $f(g)$ contains the g×g-grid as a minor. For every …
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 …
This lecture series covers several different techniques on embedding paths/trees/cycles in (pseudo)random graphs/expanders as (induced) subgraphs.
This lecture series covers several different techniques on embedding paths/trees/cycles in (pseudo)random graphs/expanders as (induced) subgraphs.