In a breakthrough discovery, mathematicians have created a Steiner triple system with arbitrary waist size, proving the Erdös conjecture after 50 years. Paul Erdös, a renowned mathematician, is known for his contributions to number theory and combinatorics. One of his combinatorial designs, the Steiner triple system, describes the number of possible triple combinations under certain conditions. While Erdös believed that such systems were possible, the math community had only been able to prove systems with a maximum of six people. However, scientists at the Institute of Science and Technology Austria (ISTA) have now proven the existence of a Steiner triple system with arbitrary waist size.
To prove the existence of this system, the scientists had to avoid algebra and instead use methods from probability theory and probabilistic combinatorics. They also utilized two new methods: retrospective analysis and sparsification. The Erdös conjecture, which was proposed 50 years ago, has now been proven. The authors of the study explain that they can prove the lower bound for the number of r-sparse Steiner triple systems with a given set of vertices. This means that seven vertices can form exactly seven triples without any duplicate pairs.
This achievement was made possible by combining methods from different sub-disciplines of mathematics. Matthew Kwan from ISTA believes that working on different problems can lead to discovering new techniques that can be applied in other areas. The breakthrough discovery of the Steiner triple system with arbitrary waist size has significant implications for computer codes and experiments. It also highlights the importance of collaboration and interdisciplinary approaches in solving complex mathematical problems.
