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Mathematicians Establish Erdős Design Idea


Fall of 1972, Vance Faber was a new professor at the University of Colorado. When two mathematicians, Paul Erdős and László Lovász, came to visit, Faber decided to have a tea party. Erdős in particular had a world-class reputation as a courageous and dynamic explorer, and Faber’s colleagues were eager to meet him.

“We’re there, as at most tea parties, Erdős sits in the corner, surrounded by his fans,” Faber said. “They talk at the same time, often in several different languages.”

Erdős, Faber, and Lovász were merely looking at discussions on hypergraphs, which was a new promise in graphic literature at the time. After arguing, they came up with one question, which later became known as the Erdős-Faber-Lovász idea. It affects the amount of colors that need to be drawn in the margins of hypergraph for some problems.

“It was the easiest thing to do,” says Faber, now a mathematician at the Institute for Defense Analyzes’ Center for Computing Science. “We worked a little bit at the party and said, ‘Oh, we can do it tomorrow.’ That never happened. ”

The situation became more serious than he had anticipated. Erdős often advertised it as one of his three favorite items, and offered a prize for the answer, which rose to $ 500 when mathematics realized the problem. The problem was well-known in the graphic categories and attracted a lot of effort to solve it, and none of it worked well.

But now, nearly 50 years later, a team of mathematicians has finally confirmed that the tea party is in harmony. Mu preprint published in January, limits the number of colors that may be required to keep the edges of other hypergraphs so that there are no overlapping parts that have the same color. He asserts that the number of colors cannot be greater than the quantity of hypergraph form.

The technique involves excluding certain parts of the graph and drawing other colors indiscriminately, combining the ideas that researchers have used in recent years to solve a number of problems that have been open for a long time. He was not seen by Erdős, Faber and Lovász when dreaming about the problem. But now, looking at his mind, the two surviving mathematicians could enjoy the math that had aroused their curiosity.

“It’s a beautiful job,” he said The Bridegroom, of Eötvös Loránd University. “I was very happy to see this happen.”

Sufficient Types

As Erdős, Faber, and Lovász drank tea and chatted, they had a new graph-like concept in their minds. Ordinary graphs are built from a location, called voices, connected by lines, called margins. Each side connects exactly two parts. But the hypergraphs Erdős, Faber and Lovász that they see are not strict: On their side you can straighten both sides.

This far-reaching concept makes hypergraphs more flexible than their talk-and-talk relatives. Ordinary graphs can simply describe the relationship between two things, such as two friends on a social network (where each person is represented by a vertex). But to define the relationship between two people — such as having a group — each side must include more than two people, which hypergraphs allow.

However, this change comes at a price: It is more difficult to confirm the type of hypergraphs than in ordinary graphs.

“Many miracles [of graph theory] it can go away or things get worse when you start using hypergraphs, ”he said Gil Kalai of IDC Herzliya and the University of Jerusalem.

For example, color problems are more difficult with hypergraphs. In this case, the goal is to draw in both sides of the graph (or hypergraph) so that no two groups meet the vertex of the same type. The smallest types required to do this are known as the chromatic index of the graph.

Erdős-Faber-Lovász ‘s idea is a question about a type of hypergraph with a slight edge. In these groups, known as parallel hypergraphs, no two parts are allowed to exceed one vertex. The hypothesis states that the chromatic index of the same hypergraph does not exceed the maximum. Alternatively, if a solid hypergraph line has nine lines, the edges can be painted with no more than nine colors, regardless of how you draw.


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