The "schoolgirl problem" was described by a mathematician when he wasn't fulfilling his main responsibility as a priest.
A hypergraph is what a modern mathematician would think of this type of problem. Each group of 15 girls is a triangle with three lines and edges connecting them.
If there is an arrangement of these triangles that connects all the schoolgirls to one another, but with the added restriction that no two triangles share an edge, what does that mean? Two girls have to walk together more than one time. Each girl has to walk with two new friends every day for a week in order for them to get together.
For the past two centuries, mathematicians have beenguiled by this problem and others. The mathematician Paul Erds posed for a picture. He wanted to know if it was possible to build a hypergraph with two different properties. The schoolgirls had to connect by one triangle. The graph is made dense by this property. The triangles must be spread out in a precise manner. For any small group of triangles, there must be at least three more than one. David Conlon is a mathematician at the California Institute of Technology.
Four mathematicians proved in January that it is possible to build a hypergraph as long as you have enough resources. Allan Lo is a mathematician at the University of Birmingham. Conlon said it was an impressive piece of work.
The research team built a system that satisfied Erds by starting with a random process for choosing triangles and engineering it to fit their needs. There are a lot of difficult modifications that go into the proof.
They built the hypergraph out of individual triangles. Imagine our 15 girls. Line up the pairs.
First, no two triangles share an edge, and second, the triangles must be traced on top of the lines. The systems that meet this requirement are called triple systems. Second, make sure that every small subset of triangles uses a sufficient number of nodes.
The researchers did this in a certain way.
You are building houses out of Lego bricks. The first few buildings you make are very fancy. Set them aside once youTrademarkiaTrademarkiaTrademarkiaTrademarkiaTrademarkiaTrademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia,Trademarkia.Trademarkia,Trademarkia,Trademarkia,Trademarkia, They will be used as anabsorber, a sort of structured stockpile.
You should start making buildings out of your bricks now. Stray bricks or homes that are not sound can be found when your Legos run out. You can use bricks out here and there since the buildings are so heavy.
You are attempting to create triangles in the case of the triple system. A collection of edges is what your absorber is made of. The edges of the absorber can be used if you can't sort the rest of the system into triangles. The absorber is broken down into triangles when you finish.
Sometimes absorption isn't always effective. The process has been tinkered with by mathematicians. A powerful variant called iterative absorption divides the edges into a nested sequence of sets so that each one acts as an absorber.
There have been huge improvements over the last decade or so. They have really carried it up to the level of high art at this point.
Even with iterative absorption, Erds' problem was hard to solve. Mehtaab Sawhney is a graduate student at the Massachusetts Institute of Technology and one of the researchers who solved the problem. There were a lot of technical tasks.
In other applications of iterative absorption, once you finish covering a set, you can forget about it. The mathematicians couldn't do that because of Erds' conditions. There is a cluster of triangles that could cause problems.
Sawhney said that a triangle you chose 500 steps ago needs to be remembered.
The four realized that if they chose their triangles carefully, they could avoid the need to keep track of everything. If you want to make sure that a set of 100 triangles is chosen with the correct probability, you should think about it.
The authors of the new paper think they can extend their technique to other problems. The Latin squares problem is a simplification of a sudoku puzzle.
There are a number of questions that could yield to absorption methods. Random processes are a really powerful tool and there are a lot of problems in the field. Since the 1960s, there has been a solution to the Ryser-Brualdi-Stein problem.
Maya Stein said that absorption has come a long way since its inception 30 years ago. It's great to see how these methods evolve.
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