Avoiding monochromatic solutions to 3-term equations

Given an equation, the integers $[n] = \{1, 2,\dotsc, n \}$ 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 of 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.

Keywords

additive combinatorics, Ramsey theory

2010 Mathematics Subject Classification

Primary 05D10. Secondary 11B75.

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Received 9 April 2022

Published 28 December 2022