Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Set Coverings and Uniformity Theorems

In course

MATH-467: Probabilistic methods in combinatorics

The 'probabilistic method' is a fundamental tool in combinatorics. The basic idea is as follows: to prove that an object (for example, graph) with certain properties exists, it suffices to prove that if the object is chosen at random, then it has the desir ...

Description

This lecture covers the concept of set coverings, where the goal is to find the smallest number of subsets needed to cover all elements of a given set. It also delves into Rödl's introduction of the uniformity theorem, which deals with the existence of certain structures in hypergraphs. The lecture further explores Lemmas related to uniform hypergraphs and their properties, providing insights into the structure of these mathematical objects.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (27)

Hypergraph

In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair , where is a set of elements called nodes, vertices, points, or elements and is a set of pairs of subsets of . Each of these pairs is called an edge or hyperedge; the vertex subset is known as its tail or domain, and as its head or codomain. The order of a hypergraph is the number of vertices in .

Matching in hypergraphs

In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph. Recall that a hypergraph H is a pair (V, E), where V is a set of vertices and E is a set of subsets of V called hyperedges. Each hyperedge may contain one or more vertices. A matching in H is a subset M of E, such that every two hyperedges e_1 and e_2 in M have an empty intersection (have no vertex in common).

Bipartite hypergraph

In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph. Property B The weakest definition of bipartiteness is also called 2-colorability. A hypergraph H = (V, E) is called 2-colorable if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge meets both X and Y. Equivalently, the vertices of H can be 2-colored so that no hyperedge is monochromatic.

Hall-type theorems for hypergraphs

In the mathematical field of graph theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessler, Ron Aharoni, Penny Haxell, Roy Meshulam, and others. Hall's marriage theorem provides a condition guaranteeing that a bipartite graph (X + Y, E) admits a perfect matching, or - more generally - a matching that saturates all vertices of Y. The condition involves the number of neighbors of subsets of Y.

Vertex cover in hypergraphs

In graph theory, a vertex cover in a hypergraph is a set of vertices, such that every hyperedge of the hypergraph contains at least one vertex of that set. It is an extension of the notion of vertex cover in a graph. An equivalent term is a hitting set: given a collection of sets, a set which intersects all sets in the collection in at least one element is called a hitting set. The equivalence can be seen by mapping the sets in the collection onto hyperedges. Another equivalent term, used more in a combinatorial context, is transversal.

Related lectures (32)

Sets and Operations: Introduction to Mathematics

Covers the basics of sets and operations in mathematics, from set properties to advanced operations.

Relations in Computer Science

Explores the properties of relations in computer science, including equivalence relations and the partition of a set.

Determinantal Point Processes and Extrapolation

Covers determinantal point processes, sine-process, and their extrapolation in different spaces.

Limits and Colimits in Functor Categories

Explores limits and colimits in functor categories, focusing on equalizers, pullbacks, and their significance in category theory.

The Nerve and Geometric Realization