In front of you, there are 100 dots. Draw lines between the points using connect-the-dots. How many lines can you make? There is a square. Is it an 11 point star?
These types of problems have been around for a long time. The answer to the question was given in a paper posted on April 26. A graph is an arrangement of dots and lines. A regular graph is the type of graph they want to find. A regular graph has the same number of lines connecting each point. There are two edges in a 2-regular graph. Each nodes has a degree of 600.
A portion of the dots and edges will eventually become regular graphs if you keep adding edges. When you try to place five edges among four nodes, they become inescapable. The problem first asked in 1975 was solved by Janzer and Sudakov.
The property is simple to state, has an old history, and has a lot of immediate consequences.
The answer to the problem of 1 or 2 regular graphs was established long before 1975. These objects are very easy to use. Any nonempty graph can be considered a 1-regular graph if it consists of two nodes and one edge. Think of a closed loop as a 2-regular graph. 3-regular graphs and higher are more complex and have different structures. Gil Kalai is a professor at Reichman University and the Hebrew University of Jerusalem.
When they first came up with their problem, they had some ideas about the 3-regular situation. They were aware of graphs that didn't have regular substructures. They knew that a graph with more than one edge had to have a regular structure inside. The right threshold had to be between order n and order n 8/6, where "order n" just means "multiply by some constant" There were still many possibilities.
Lszl Pyber narrowed the options considerably in 1985 when he showed that the answer had to be less than order nlog. When a graph has a lot of points, mathematicians care about what happens. If n is larger than nlog, then n 8/6 is larger than nlog.
After that, Pyber, Vojtch Rdl, and Endre Szemerédi created a graph with an order nlog edge that did not have a regular graph with degree 3 or more. The answer to the problem ranged from the low end to the high end.
The question was stagnant for almost four decades until this March when the two men decided to try the problem. This is a high-risk, high-reward problem. Sometimes you need to try if you don't solve it.
The pair looked into Pyber, Rdl, and Szemerédi's paper in order to figure out how they avoided producing a regular graph. Some of the edges were connected to an enormous number of the other edges. It wasn't possible to get a more regular substructure from these different types of nodes.
Jacques Verstraete, a mathematician at the University of California, San Diego, said that the construction inspired him.
If you add more edges, the whole picture will change. Regular graphs start to show up suddenly. The two authors were able to find a regular graph from all types of graphs because of that feature.
They were able to prove that the order nlog(log(n)) edges will have a regular graph. If you are looking for a 3-regular graph or a million-regular graph, the same order of the solution is used. The long-ago result means the order can't be made any smaller. Verstraete said it was remarkable that they had solved the problem for more than three decades.
The two men have found ways to apply their knowledge to other problems. The mathematician Carsten Thomassen wrote a problem about graphs in 1983. Thomassen wanted to take a graph and extract a substructure that avoids short loops of edges, as well as making sure each nodes is attached to a number of edges. If the original graph has enough edges to start with, the exercise can be done. They used their work to answer a question from 1970 that was not related to the Erds-Sauer problem.
The paper gives a coda to a symphony. He said that it uses all of the work done by the previous guys. Everybody who thought about the problem thought in the right way.
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