기초과학연구원 이산수학그룹
A family $\mathcal F$ of subsets of is called s-saturated if it contains no s pairwise disjoint sets, and moreover, no set can be added to $\mathcal F$ while preserving this property. More than 40 years ago, Erdős and Kleitman conjectured that an s-saturated family of subsets of has size at least $(1 – 2^{-(s-1)})2^n$. …
A hypergraph is linear if every pair of two distinct edges shares at most one vertex. A longstanding conjecture by Erdős, Faber, and Lovász in 1972, states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this talk, I will present the ideas to prove the conjecture for …
In the "cake partition" problem n players have each a list of preferred parts for any partition of the interval ("cake") into n sub-intervals. Woodall, Stromquist and Gale proved independently that under mild conditions on the list of preferences (like continuity) there is always a partition and assignment of parts to the players, in which every player gets …
Main purpose of this talk is to introduce a connection between restriction estimates for cones and point-sphere incidence theorems in the finite field setting. First, we review the finite field restriction problem for cones and address new results on the conical restriction problems. In particular, we establish the restriction conjecture for the cone in four …
We prove a conjecture of Boros, Caro, Furedi and Yuster on the maximum number of edges in a 2-connected graph without repeated cycle lengths, which is a restricted version of a longstanding problem of Erdos. Our proof together with the matched lower bound construction of Boros, Caro, Furedi and Yuster show that this problem can …
Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say). But concerning continuous data, already real numbers naturally suggest various encodings with very different computational properties. …
This talk considers the following question at the intersection of extremal and structural graph theory: What is the maximum number of copies of a fixed forest $T$ in an $n$-vertex graph in a graph class $\mathcal{G}$ as $n\to \infty$? I will answer this question for a variety of sparse graph classes $\mathcal{G}$. In particular, we show that the answer is …
We prove that if $n \geq 3$, then any family of $3n-3$ sets of matchings of size $n$ in any graph has a rainbow matching of size $n$. This improves on a previous result, in which $3n-3$ is replaced by $3n-2$. We also prove a "cooperative" generalization: for $t > 0$ and $n \geq 3$, …
Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing …
Graph saturation is one of the oldest areas of investigation in extremal combinatorics. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. The function $\operatorname{sat}(n,F)$ is defined to be the minimum number of edges in an $n$-vertex …