On the finiteness of some -color Rado numbers

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• 作者：S.D. Adhikari^a ; ^{adhikari@hri.res.in} ; L. Boza^b ; ^boza@us.es ; S. Eliahou^c ; ^d ; ^{eliahou@lmpa.univ-littoral.fr} ; J.M. Marí ; n^e ; ^jmarin@us.es ; M.P. Revuelta^e ; ^{pastora@us.es} ; M.I. Sanz^e ; ^isanz@us.es
• 关键词：Schur numbers ; Sum-free sets ; Rado numbers ; Extremal problem ; SAT problem ; Hypergraph coloring ; Partition-regular equation
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 February 2017
• 年：2017
• 卷：340
• 期：2
• 页码：39-45
• 全文大小：451 K
• 卷排序：340

文摘

For integers k,n,ck,n,c with k,n≥1k,n≥1, the nn-color Rado number Rk(n,c)Rk(n,c) is defined to be the least integer NN if any, or infinity otherwise, such that for every nn-coloring of the set {1,2,…,N}{1,2,…,N}, there exists a monochromatic solution in that set to the linear equation x1+x2+⋯+xk+c=xk+1.A recent conjecture of ours states that Rk(n,c)Rk(n,c) should be finite if and only if every divisor d≤nd≤n of k−1k−1 also divides cc. In this paper, we complete the verification of this conjecture for all k≤7k≤7. As a key tool, we first prove a general result concerning the degree of regularity over subsets of ZZ of some linear Diophantine equations.