Humpty Dumpty had all the king's horses and all the king's men, but it wasn't enough to put him back together again. Which begs the question, how many horses and men did he actually need?
That question — how much is enough to guarantee success — gets at the heart of new research by a team of mathematicians including FIU Ph.D. student Justin Wisby. They are studying Ramsey theory. It is an area of extremal graph theory, mathematics that searches for the minimum or maximum needed to guarantee a particular outcome. Ramsey theory focuses specifically on patterns and the point at which they become impossible to avoid.
"The idea being that if a task is possible to accomplish, can we do it with less," Wisby said. "Or in the case of Humpty Dumpty, how many more men would have been required to save the egg? That's the extremal part, finding the extremes of the set where a task is possible."
The research is theoretical, but the mathematics behind finding these kinds of limits has broader applications. Extremal graph theory contributes to discrete optimization — an area of mathematics commonly relied on by companies to make core decisions that could save money, reduce waste, improve scheduling and other efficiencies. Think of how FedEx determines the best routes for delivery trucks, how airlines assign planes to gates, how Nike manages its inventory — most likely discrete optimization. Research like Wisby's works on the same idea that all these problems share — regardless of how complicated, how much is enough to guarantee a solution.
The research uses a mathematical game with two opponents: builder, who connects points trying to create a specific pattern on an ordered line, and painter, who colors those connections in an attempt to prevent the pattern. Inevitably, builder can force the pattern to appear. The question is: What is the minimum number of moves needed to guarantee the desired outcome of the pattern?
Using a game involving pattern building, the researchers established new mathematical boundaries that narrow down that answer for certain types of patterns. Their work chips away at larger questions in Ramsey theory, which Wisby says includes some of the most difficult unsolved problems in mathematics.
The project began at the 2023 Graduate Research Workshop in Combinatorics in Wyoming, where Wisby joined researchers he had never worked with before.
"Math will never run out of research problems, and this project was a great way for me to spread my wings and collaborate with new people on a project they were passionate about," he said.
While Humpty Dumpty may have been beyond saving with the knowledge of the 1800s, today's mathematics could provide the answer to just how many horses and men he needed. With those precise boundaries of success, Wisby says no puzzle, complex schedule, resource challenge or corporate logistic is beyond a solution.
