7 papers
An elementary proof of the 2-regularity of Pythagorean triples
William J. Wesley
We give a new proof of the fact that every 2-coloring of the positive integers contains a monochromatic Pythagorean triple. This result was originally proven by Heule, Kullmann, an…
The Fibonacci numbers are not 3-accessible
William J. Wesley
A -diffsequence is a sequence of integers such that for . The set is called -accessible if every -coloring of…
Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment
Stephen Chung, Wenyu Du, William J. Wesley
We study autonomous mathematical discovery in the Station, an open-world multi-agent environment in which AI agents from different model families pursue a shared research goal with…
Optimization Tools for Computing Colorings of with Few Monochromatic Solutions on -variable Linear Equations
Jesús A. De Loera, Denae Ventura, Liuyue Wang +1
A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations there is a threshold value (the Rado number of ) such that for any -c…
Improved Ramsey-type theorems for Fibonacci numbers and other sequences
William J. Wesley
Van der Waerden's theorem states that for any positive integers and , there exists a smallest value , called the van der Waerden number, such that every -colo…
Rado Numbers and SAT Computations
Yuan Chang, Jesús A. De Loera, William J. Wesley
Given a linear equation , the -color Rado number is the smallest integer such that every -coloring of contains a monoc…
Ramsey Numbers through the Lenses of Polynomial Ideals and Nullstellensätze
Jesús A. De Loera, William J. Wesley
In this article we study the Ramsey numbers through Hilbert's Nullstellensatz and Alon's Combinatorial Nullstellensatz. We give polynomial encodings whose solutions corres…