2023/06/06 by Gaiser, Collier
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.04029
Let fr(k) be the smallest positive integer n such that every r-coloring of \1,2,...,n\ has a monochromatic solution to the nonlinear equation 1/x1+⋯+1/xk=1/y, where x1,...,xk are not necessarily distinct. Brown and Rödl [Bull. Aust. Math. Soc. 43(1991): 387-392] proved that f2(k)=O(k6). In this paper, we prove that f2(k)=O(k3). The main ingredient in our proof is a finite set A⊆ℕ such that every 2-coloring of A has a monochromatic solution to the linear equation x1+⋯+xk=y and the least common multiple of A is sufficiently small. This approach can also be used to study fr(k) with r>2. For example, a recent result of Boza, Marín, Revuelta, and Sanz [Discrete Appl. Math. 263(2019): 59-68] implies that f3(k)=O(k43).