On monochromatic solutions to linear equations over the integers

2024/10/17 by Dong, Dingding, Mani, Nitya, Pham, Huy Tuan +1
#05D40 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.13758

Abstract

We study the number of monochromatic solutions to linear equations in a 2-coloring of \1,…,n\. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any 2-coloring of \1,…,n\. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over \mathbbF_p, the four-term equation x₁ + 2x₂ - x₃ - 2x₄ = 0 is uncommon over \1,…,n\.

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