2022/03/05 by Yang, Gang, Mao, Yaping, He, Changxiang +1
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2203.04126
The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. Let E be a linear equation. Denote by Rr(E) the minimal integer, if it exists, such that any r-coloring of [1,Rr(E)] must admit a monochromatic solution to E. In this paper, we give upper and lower bounds for the Rado number of ∑i=1m-2xi+kxm-1=ℓ xm, and some exact values are also given. Furthermore, we derive some results for the cases that ℓ=m=4 and m=5, ℓ=k+i (1≤ i≤ 5). As a generalization, the r-color Rado numbers for linear equations E1,E2,...,Er is defined as the minimal integer, if it exists, such that any r-coloring of [1,Rr(E1,E2,...,Er)] must admit a monochromatic solution to some Ei, where 1≤ i≤ r. A lower bound for Rr(E1,E2,...,Er) and the exact values of R2(x+y=z,ℓ x=y)=5k and R2(x+y=z, x+a=y) was given by Lovász Local Lemma.