기초과학연구원 이산수학그룹
In 1975, Erdős asked the following question: what is the smallest function $f(n)$ for which all graphs with $n$ vertices and $f(n)$ edges contain two edge-disjoint cycles $C_1$ and $C_2$, such that the vertex set of $C_2$ is a subset of the vertex set of $C_1$ and their cyclic orderings of the vertices respect each …
The twin-width of a graph G can be defined as the least integer d such that there is a sequence of length |V(G)| of (strictly) coarser and coarser partitions of its vertex set V(G), and every part X of every partition P of the sequence has at most d other parts Y of P with …
We show that that the maximum number of of edges in a $3$-uniform hypergraph without a Berge-cycle of length four is at most $(1+o(1)) \frac{n^{3/2}}{\sqrt{10}}$. This improves earlier estimates by Győri and Lemons and by Füredi and Özkahya. Joint work with Ergemlidze, Győri, Methuku, Salia.
A pure pair in a graph G is a pair of subsets A, B of the vertex set of G such that in G, either all of the edges or none of the edges between A and B are present. Pure pairs have been studied recently motivated by their connections to the Erdos-Hajnal conjecture. In …
In this talk we will have a brief introduction to oriented matroids and their relation to real-representability.
For a graph G and an integer d, the dth power of G is the graph $G^d$ on the same vertex set as G where two vertices are considered adjacent if and only if they are at distance at most d in G. Assuming that G is sparse, what can we say about the structure …
Ordered Ramsey numbers were introduced in 2014 by Conlon, Fox, Lee, and Sudakov. Their results included upper bounds for general graphs and lower bounds showing separation from classical Ramsey numbers. We show the first nontrivial results for ordered Ramsey numbers of specific small graphs. In particular we prove upper bounds for orderings of graphs on four vertices, …
A graph $G$ is semilinear of bounded complexity if the vertices of $G$ are elements of $\mathbb{R}^{d}$, and the edges of $G$ are defined by the sign patterns of $t$ linear functions, where $d$ and $t$ are constants. In this talk, I will present several results about the symmetric and asymmetric Ramsey properties of semilinear …
Bouchet introduced isotropic systems in 1983 unifying some combinatorial features of binary matroids and 4-regular graphs. The concept of isotropic system is a useful tool to study vertex-minors of graphs and yet it is not well known. I will give an introduction to isotropic systems.
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that typically occur together. They can thereby discover, relate, and structure types: of behaviour, political views, texts, or proteins. Tangles offer a new, quantitative, paradigm for grouping phenomena rather than things. They can identify key phenomena …