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 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, the deputy director of the Center for Mathematical Modeling at the University of Chile, said that absorption has come a long way since it was first conceived. It's great to see how these methods evolve.

The original story was originally published in the journal of the Simons Foundation, an editorially independent publication that covers research developments and trends in mathematics and the physical and life sciences.