Saturday 22 March 2025
Computer scientists have long been fascinated by a problem known as Ramsey theory, which deals with how often certain patterns appear in large groups of things. For instance, if you were to take a group of 28 people and ask them to sit down, it’s likely that some group of three would contain at least one person who knows another person in the group. This is because it’s difficult to find a way for all 27 pairs of people to be complete strangers.
Recently, researchers have been using computer algorithms to help solve problems like this, and they’ve made significant progress. One approach they’re taking is called SAT+CAS, which combines two techniques: satisfiability solving (SAT) and computer algebra systems (CAS).
SAT is a way for computers to figure out if a given set of rules can be satisfied simultaneously. It’s like trying to find the right combination of numbers that will make a mathematical equation true. In this case, the researchers are using SAT to search for patterns in large groups.
Computer algebra systems, on the other hand, are designed to help with complex mathematical calculations. They can simplify equations, solve polynomials, and even prove mathematical theorems.
By combining these two approaches, the researchers have been able to tackle problems that were previously thought to be too difficult to solve. One of their recent successes was solving a problem known as R(3,8), which deals with how often certain patterns appear in groups of 28 people.
To do this, they created a computer program that used SAT to search for possible solutions, and then used CAS to verify those solutions and eliminate any incorrect ones. The result was a proof that R(3,8) equals 28 – in other words, it’s impossible to find a group of 27 or fewer people where no one knows anyone else.
This breakthrough has significant implications for fields like graph theory, which is the study of how nodes (like people) connect to each other. It also opens up new possibilities for solving other problems that were previously thought to be unsolvable.
The researchers are now working on applying this approach to even more complex problems. For instance, they’re trying to solve R(3,9), which deals with groups of 36 people and is much harder to solve than R(3,8).
Their ultimate goal is to develop a way for computers to automatically prove mathematical theorems, without needing human intervention.
Cite this article: “Unlocking Mathematical Secrets with SAT+CAS”, The Science Archive, 2025.
Ramsey Theory, Computer Science, Sat, Cas, Satisfiability Solving, Computer Algebra Systems, Graph Theory, Pattern Recognition, Mathematical Equations, Theorem Proving.
