2021/03/04 by Kevin Costello, Costello, Kevin P., Gabriel Elvin +1
Computer Science · Mathematics · #05D10 #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2103.03350
openalex publication_date 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an equation, the integers [n] = \1, 2, …, n\ as inputs, and the colors red and blue, how can we color [n] in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is only known for a handful of equations, but much progress has been made on improving upper and lower bounds on minima for various equations. A well-studied characteristic an equation, which has its roots in graph Ramsey theory, is to determine if the minimum number of monochromatic solutions can be achieved (asymptotically) by uniformly random colorings. Such equations are called common. We prove that no 3-term equations are common and provide a lower bound for a specific class of 3-term equations.