Schur-like numbers and a lemma of Shearer

2025/07/29 by Kosciuszko, Tomasz
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.21656

Abstract

Suppose that each number 1,2,...,N has one of n colours assigned. We show that if there are no monochromatic solutions to the equation x₁+x₂+x₃=y₁+y₂, then N=O((n!)^1/2), improving upon a result of Cwalina and Schoen. Further, a stronger bound of N=O(((n-k)!)^1/2), where k≫(log n)/(loglog n) is shown for colourings avoiding solutions to the equation x₁+x₂+...+x₁₂=y₁+y₂+...+y₉. Finally, some remarks on other equations are presented.

Citations

• Improved bounds for five-term arithmetic progressions
• Strong Bounds for 3-Progressions

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